Download Scattering, Absorption, and Emission of Light by Small Particles

Scattering, Absorption, and Emission of Light by
Small Particles
This volume provides a thorough and up-to-date treatment of electromagnetic scattering
by small particles. First, the general formalism of scattering, absorption, and emission of
light and other electromagnetic radiation by arbitrarily shaped and arbitrarily oriented
particles is introduced, and the relation of radiative transfer theory to single-scattering
solutions of Maxwell’s equations is discussed. Then exact theoretical methods and
computer codes for calculating scattering, absorption, and emission properties of
arbitrarily shaped particles are described in detail. Further chapters demonstrate how the
scattering and absorption characteristics of small particles depend on particle size,
refractive index, shape, and orientation. The work illustrates how the high efficiency and
accuracy of existing theoretical and experimental techniques and the availability of fast
scientific workstations result in advanced physically based applications of electromagnetic
scattering to noninvasive particle characterization and remote sensing. This book will be
valuable for science professionals, engineers, and graduate students in a wide range of
disciplines including optics, electromagnetics, remote sensing, climate research, and
biomedicine.
MICHAEL I. MISHCHENKO is a Senior Scientist at the NASA Goddard Institute for
Space Studies in New York City. After gaining a Ph.D. in physics in 1987, he has been
principal investigator on several NASA and DoD projects and has served as topical editor
and editorial board member of leading scientific journals such as Applied Optics, Journal
of Quantitative Spectroscopy and Radiative Transfer, Journal of the Atmospheric
Sciences, Waves in Random Media, Journal of Electromagnetic Waves and Applications,
and Kinematics and Physics of Celestial Bodies. Dr. MISHCHENKO is a recipient of the
Henry G. Houghton Award of the American Meteorological Society, a Fellow of the
American Geophysical Union, a Fellow of the Optical Society of America, and a Fellow
of The Institute of Physics. His research interests include electromagnetic scattering,
radiative transfer in planetary atmospheres and particulate surfaces, and remote sensing.
LARRY D. TRAVIS is presently Associate Chief of the NASA Goddard Institute for Space
Studies. He gained a Ph.D. in astronomy at Pennsylvania State University in 1971. Dr.
TRAVIS has acted as principal investigator on several NASA projects and was awarded a
NASA Exceptional Scientific Achievement Medal. His research interests include the
theoretical interpretation of remote sensing measurements of polarization, planetary
atmospheres, atmospheric dynamics, and radiative transfer.
ANDREW A. LACIS is a Senior Scientist at the NASA Goddard Institute for Space
Studies, and teaches radiative transfer at Columbia University. He gained a Ph.D. in
physics at the University of Iowa in 1970 and has acted as principal investigator on
numerous NASA and DoE projects. His research interests include radiative transfer in
planetary atmospheres, the absorption of solar radiation by the Earth’s atmosphere, and
climate modeling.
M. I. MISHCHENKO and L. D. TRAVIS co-edited a monograph on Light Scattering by
Nonspherical Particles: Theory, Measurements, and Applications published in 2000 by
Academic Press.
Revised electronic edition
Michael I. Mishchenko
Larry D. Travis
Andrew A. Lacis
NASA Goddard Institute for Space
pace Studies, New York
The first hardcopy edition of this book was published in 2002 by
CAMBRIDGE UNIVERSITY PRESS
The Edinburgh Building
Cambridge CB2 2RU
UK
http://www.cambridge.org
A catalogue record for this book is available from the British Library
ISBN 0 521 78252 X hardback
© NASA 2002
The first electronic edition of this book was published in 2004 by
NASA Goddard Institute for Space Studies
2880 Broadway
New York, NY 10025
USA
http://www.giss.nasa.gov
The electronic edition is available at the following Internet site:
http://www.giss.nasa.gov/~crmim/books.html
This book is in copyright, except in the jurisdictional territory of the
United States of America. The moral rights of the authors have been
asserted. Single copies of the book may be printed from the Internet site
http://www.giss.nasa.gov/~crmim/books.html for personal use as allowed
by national copyright laws. Unless expressly permitted by law, no
reproduction of any part may take place without the written permission
of NASA.
Contents
Preface to the electronic edition xi
Preface to the original hardcopy edition xiii
Acknowledgments xvii
Part I
Basic Theory of Electromagnetic Scattering, Absorption, and
Emission 1
Chapter 1
Polarization characteristics of electromagnetic radiation 8
1.1
1.2
1.3
1.4
1.5
1.6
Maxwell’s equations, time-harmonic fields, and the Poynting vector 8
Plane-wave solution 12
Coherency matrix and Stokes parameters 15
Ellipsometric interpretation of Stokes parameters 19
Rotation transformation rule for Stokes parameters 24
Quasi-monochromatic light and incoherent addition of Stokes
parameters 26
Further reading 30
Chapter 2
Scattering, absorption, and emission of electromagnetic
radiation by an arbitrary finite particle 31
2.1
2.2
2.3
2.4
2.5
Volume integral equation 31
Scattering in the far-field zone 35
Reciprocity 38
Reference frames and particle orientation 42
Poynting vector of the total field 46
v
Scattering, Absorption, and Emission of Light by Small Particles
vi
2.6
2.7
2.8
2.9
2.10
2.11
Phase matrix 49
Extinction matrix 54
Extinction, scattering, and absorption cross sections 56
Radiation pressure and radiation torque 60
Thermal emission 63
Translations of the origin 66
Further reading 67
Chapter 3
Scattering, absorption, and emission by collections of
independent particles 68
3.1
Single scattering, absorption, and emission by a small volume
element comprising randomly and sparsely distributed particles 68
Ensemble averaging 72
Condition of independent scattering 74
Radiative transfer equation and coherent backscattering 74
Further reading 82
3.2
3.3
3.4
Chapter 4
Scattering matrix and macroscopically isotropic and
mirror-symmetric scattering media 83
4.1
4.2
Symmetries of the Stokes scattering matrix 84
Macroscopically isotropic and mirror-symmetric scattering
medium 87
Phase matrix 88
Forward-scattering direction and extinction matrix 91
Backward scattering 94
Scattering cross section, asymmetry parameter, and radiation
pressure 95
Thermal emission 97
Spherically symmetric particles 98
Effects of nonsphericity and orientation 99
Normalized scattering and phase matrices 100
Expansion in generalized spherical functions 103
Circular-polarization representation 105
Radiative transfer equation 108
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
Part II
Calculation and Measurement of Scattering and
Absorption Characteristics of Small Particles 111
Chapter 5
T-matrix method and Lorenz–Mie theory 115
5.1
5.2
T-matrix ansatz 116
General properties of the T matrix 119
Rotation transformation rule 119
5.2.1
Contents
5.2.2
5.2.3
5.2.4
5.3
5.4
5.5
5.6
5.7
5.8
5.8.1
5.8.2
5.8.3
5.8.4
5.8.5
5.9
5.10
5.10.1
5.10.2
5.10.3
5.10.4
5.11
5.11.1
5.11.2
5.11.3
5.11.4
5.11.5
5.11.6
5.11.7
5.12
5.13
Symmetry relations 121
Unitarity 122
Translation transformation rule 125
Extinction matrix for axially oriented particles 127
Extinction cross section for randomly oriented particles 132
Scattering matrix for randomly oriented particles 133
Scattering cross section for randomly oriented particles 138
Spherically symmetric scatterers (Lorenz–Mie theory) 139
Extended boundary condition method 142
General formulation 142
Scale invariance rule 147
Rotationally symmetric particles 148
Convergence 150
Lorenz–Mie coefficients 153
Aggregated and composite particles 154
Lorenz–Mie code for homogeneous polydisperse spheres 158
Practical considerations 158
Input parameters of the Lorenz–Mie code 162
Output information 163
Additional comments and illustrative example 164
T-matrix code for polydisperse, randomly oriented, homogeneous,
rotationally symmetric particles 165
Computation of the T matrix for an individual particle 167
Particle shapes and sizes 171
Orientation and size averaging 172
Input parameters of the code 173
Output information 175
Additional comments and recipes 176
Illustrative examples 178
T-matrix code for a homogeneous, rotationally symmetric particle
in an arbitrary orientation 180
Superposition T-matrix code for randomly oriented two-sphere
clusters 186
Further reading 189
Chapter 6
Miscellaneous exact techniques 191
6.1
6.2
6.3
6.4
6.5
6.6
Separation of variables method for spheroids 192
Finite-element method 193
Finite-difference time-domain method 195
Point-matching method 196
Integral equation methods 197
Superposition method for compounded spheres and spheroids 201
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Scattering, Absorption, and Emission of Light by Small Particles
6.7
Comparison of methods, benchmark results, and computer codes 202
Further reading 205
Chapter 7
Approximations 206
7.1
7.2
7.3
7.4
7.5
7.6
Rayleigh approximation 206
Rayleigh–Gans approximation 209
Anomalous diffraction approximation 210
Geometrical optics approximation 210
Perturbation theories 221
Other approximations 222
Further reading 223
Chapter 8
Measurement techniques 224
8.1
8.2
Measurements in the visible and infrared 224
Microwave measurements 230
Part III
Scattering and Absorption Properties of Small
Particles and Illustrative Applications 235
Chapter 9
Scattering and absorption properties of spherical particles 238
9.1
9.2
9.3
Monodisperse spheres 238
Effects of averaging over sizes 250
Optical cross sections, single-scattering albedo, and asymmetry
parameter 252
Phase function a1(Θ ) 258
Backscattering 267
Other elements of the scattering matrix 271
Optical characterization of spherical particles 273
Further reading 278
9.4
9.5
9.6
9.7
Chapter 10 Scattering and absorption properties of nonspherical
particles 279
10.1
10.2
10.3
10.4
10.5
Interference and resonance structure of scattering patterns for
nonspherical particles in a fixed orientation; the effects of
orientation and size averaging 279
Randomly oriented, polydisperse spheroids with moderate aspect
ratios 282
Randomly oriented, polydisperse circular cylinders with moderate
aspect ratios 299
Randomly oriented spheroids and circular cylinders with extreme
aspect ratios 307
Chebyshev particles 319
Contents
10.6
10.7
10.8
10.9
10.10
10.11
Regular polyhedral particles 320
Irregular particles 322
Statistical approach 334
Clusters of spheres 337
Particles with multiple inclusions 347
Optical characterization of nonspherical particles 350
Further reading 359
Appendix A Spherical wave expansion of a plane wave in the far-field zone 360
Appendix B Wigner functions, Jacobi polynomials, and generalized spherical
functions 362
Appendix C Scalar and vector spherical wave functions 370
Appendix D Clebsch–Gordan coefficients and Wigner 3j symbols 380
Appendix E Système International units 384
Abbreviations and symbols 385
References 396
Index 441
Color plate section 449
ix
Preface to the electronic edition
This book was originally published by Cambridge University Press in June of 2002.
The entire print run was sold out in less than 16 months, and the book has been officially out of print since October of 2003. By agreement with Cambridge University
Press, this electronic edition is intended to make the book continually available via
the Internet at the World Wide Web site
http://www.giss.nasa.gov/~crmim/books.html
No significant revision of the text has been attempted; the pagination and the numbering of equations follow those of the original hardcopy edition. However, almost all
illustrations have been improved, several typos have been corrected, some minor improvements of the text have been made, and a few recent references have been added.
We express sincere gratitude to Andrew Mishchenko for excellent typesetting and
copy-editing work and to Nadia Zakharova and Lilly Del Valle for help with graphics.
The preparation of this electronic edition was sponsored by the NASA Radiation Sciences Program managed by Donald Anderson.
We would greatly appreciate being informed of any typos and/or factual inaccuracies that you may find either in the original hardcopy edition of the book or in this
electronic release. Please communicate them to Michael Mishchenko at
[email protected]
Michael I. Mishchenko
Larry D. Travis
Andrew A. Lacis
New York
May 2004
xi
Preface to the original hardcopy edition
The phenomena of scattering, absorption, and emission of light and other electromagnetic radiation by small particles are ubiquitous and, therefore, central to many science
and engineering disciplines. Sunlight incident on the earth’s atmosphere is scattered by
gas molecules and suspended particles, giving rise to blue skies, white clouds, and various optical displays such as rainbows, coronae, glories, and halos. By scattering and
absorbing the incident solar radiation and the radiation emitted by the underlying surface,
cloud and aerosol particles affect the earth’s radiation budget. The strong dependence of
the scattering interaction on particle size, shape, and refractive index makes measurements of electromagnetic scattering a powerful noninvasive means of particle characterization in terrestrial and planetary remote sensing, biomedicine, engineering, and astrophysics. Meaningful interpretation of laboratory and field measurements and remote
sensing observations and the widespread need for calculations of reflection, transmission,
and emission properties of various particulate media require an understanding of the underlying physics and accurate quantitative knowledge of the electromagnetic interaction
as a function of particle physical parameters.
This volume is intended to provide a thorough updated treatment of electromagnetic scattering, absorption, and emission by small particles. Specifically, the book
●
●
●
introduces a general formalism for the scattering, absorption, and emission of
light and other electromagnetic radiation by arbitrarily shaped and arbitrarily oriented particles;
discusses the relation of radiative transfer theory to single-scattering solutions of
Maxwell’s equations;
describes exact theoretical methods and computer codes for calculating the scat-
xiii
Scattering, Absorption, and Emission of Light by Small Particles
xiv
●
●
tering, absorption, and emission properties of arbitrarily shaped small particles;
demonstrates how the scattering and absorption characteristics of small particles
depend on particle size, refractive index, shape, and orientation; and
illustrates how the high efficiency and accuracy of existing theoretical and experimental techniques and the availability of fast scientific workstations can result in advanced physically based applications.
The book is intended for science professionals, engineers, and graduate students
working or specializing in a wide range of disciplines: optics, electromagnetics, optical and electrical engineering, biomedical optics, atmospheric radiation and remote
sensing, climate research, radar meteorology, planetary physics, oceanography, and
astrophysics. We assume that the reader is familiar with the fundamentals of classical
electromagnetics, optics, and vector calculus. Otherwise the book is sufficiently selfcontained and provides explicit derivations of all important results. Although not
formally a textbook, this volume can be a useful supplement to relevant graduate
courses.
The literature on electromagnetic scattering is notorious for discrepancies and inconsistencies in the definition and usage of terms. Among the commonly encountered differences are the use of right-handed as opposed to left-handed coordinate
systems, the use of the time-harmonic factor exp(−iω t ) versus exp(iω t ), and the
way an angle of rotation is defined. Because we extensively employ mathematical
techniques of the quantum theory of angular momentum and because we wanted to
make the book self-consistent, we use throughout only right-handed (spherical) coordinate systems and always consider an angle of rotation positive if the rotation is performed in the clockwise direction when one is looking in the positive direction of the
rotation axis (or in the direction of light propagation). Also, we adopt the timeharmonic factor exp(−iω t ), which seems to be the preferred choice in the majority of
publications and implies a non-negative imaginary part of the relative refractive index.
Because the subject of electromagnetic scattering crosses the boundaries between
many disciplines, it was very difficult to develop a clear and unambiguous notation
system. In many cases we found that the conventional symbol for a quantity in one
discipline was the same as the conventional symbol for a different quantity in another
discipline. Although we have made an effort to reconcile tradition and simplicity
with the desire of having a unique symbol for every variable, some symbols ultimately adopted for the book still represent more than one variable. We hope, however, that the meaning of all symbols is clear from the context. We denote vectors
using the Times bold font and matrices using the Arial bold or bold italic font. Unit
vectors are denoted by a caret, whereas tensors and dyadics are denoted by the dyadic
symbol ↔. The Times italic font is usually reserved for scalar variables. However,
the square root of minus one, the base of natural logarithms, and the differential sign
are denoted by Times roman (upright) characters i, e, and d, respectively. A table
Preface
xv
containing the symbols used, their meaning and dimension, and the section where
they first appear is provided at the end of the book, to assist the reader.
We have not attempted to compile a comprehensive list of relevant publications
and often cite a book or a review article where further references can be found. In
this regard, two books deserve to be mentioned specifically. The monograph by
Kerker (1969) provides a list of nearly a thousand papers on light scattering published
prior to 1970, while the recent book edited by Mishchenko et al. (2000a) lists nearly
1400 publications on all aspects of electromagnetic scattering by nonspherical and
heterogeneous particles.
We provide references to many relevant computer programs developed by various
research groups and individuals, including ourselves, and made publicly available
through the Internet. Easy accessibility of these programs can be beneficial both to
individuals who are mostly interested in applications and to those looking for sources
of benchmark results for testing their own codes. Although the majority of these programs have been extensively tested and are expected to generate reliable results in
most cases provided that they are used as instructed, it is not inconceivable that some
of them contain errors or idiosyncrasies. Furthermore, input parameters can be used
that are outside the range of values for which results can be computed accurately. For
these reasons the authors of this book and the publisher disclaim all liability for any
damage that may result from the use of the programs. In addition, although the publisher and the authors have used their best endeavors to ensure that the URLs for the
external websites referred to in this book are correct and active at the time of going to
press, the publisher and the authors have no responsibility for the websites and can
make no guarantee that a site will remain live or that the content is or will remain
appropriate.
Michael I. Mishchenko
Larry D. Travis
Andrew A. Lacis
New York
November 2001
Acknowledgments
Our efforts to understand electromagnetic scattering and its role in remote sensing and
atmospheric radiation better have been generously funded over the years by research
grants from the United States Government. We thankfully acknowledge the continuing support from the NASA Earth Observing System Program and the Department of
Energy Atmospheric Radiation Measurement Program. The preparation of this book
was sponsored by a grant from the NASA Radiation Sciences Program managed by
Donald Anderson.
We have greatly benefited from extensive discussions with Oleg Bugaenko, Brian
Cairns, Barbara Carlson, Helmut Domke, Kirk Fuller, James Hansen, Joop Hovenier,
Vsevolod Ivanov, Kuo-Nan Liou, Kari Lumme, Andreas Macke, Daniel Mackowski,
Alexander Morozhenko, William Rossow, Kenneth Sassen, Cornelis van der Mee,
Bart van Tiggelen, Gorden Videen, Tõnu Viik, Hester Volten, Ping Yang, Edgard
Yanovitskij, and many other colleagues.
We thank Cornelis van der Mee and Joop Hovenier for numerous comments
which resulted in a much improved manuscript. Our computer codes have benefited
from comments and suggestions made by Michael Wolff, Raphael Ruppin, and many
other individuals using the codes in their research. We thank Lilly Del Valle for contributing excellent drawings and Zoe Wai and Josefina Mora for helping to find papers and books that were not readily accessible.
We acknowledge with many thanks the fine cooperation that we received from the
staff of Cambridge University Press. We are grateful to Matt Lloyd, Jacqueline Garget, and Jane Aldhouse for their patience, encouragement, and help and to Susan
Parkinson for careful copy-editing work.
Our gratitude is deepest, however, to Nadia Zakharova who provided invaluable
xvii
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Scattering, Absorption, and Emission of Light by Small Particles
assistance at all stages of preparing this book and contributed many numerical results
and almost all the computer graphics.
Part I
Basic Theory of Electromagnetic Scattering,
Absorption, and Emission
A parallel monochromatic beam of light propagates in a vacuum without any change
in its intensity or polarization state. However, interposing a small particle into the
beam, as illustrated in panel (a) of the diagram on the next page, causes several distinct effects. First, the particle may convert some of the energy contained in the beam
into other forms of energy such as heat. This phenomenon is called absorption. Second, it extracts some of the incident energy and scatters it in all directions at the frequency of the incident beam. This phenomenon is called elastic scattering and, in
general, gives rise to light with a polarization state different from that of the incident
beam. As a result of absorption and scattering, the energy of the incident beam is reduced by an amount equal to the sum of the absorbed and scattered energy. This reduction is called extinction. The extinction rates for different polarization components
of the incident beam can be different. This phenomenon is called dichroism and may
cause a change in the polarization state of the beam after it passes the particle. In addition, if the absolute temperature of the particle is not equal to zero, then the particle
also emits radiation in all directions and at all frequencies, the distribution by frequency being dependent on the temperature. This phenomenon is called thermal emission.
In electromagnetic terms, the parallel monochromatic beam of light is an oscillating plane electromagnetic wave, whereas the particle is an aggregation of a large
number of discrete elementary electric charges. The oscillating electromagnetic field
of the incident wave excites the charges to oscillate with the same frequency and
thereby radiate secondary electromagnetic waves. The superposition of the secondary
waves gives the total elastically scattered field. If the particle is absorbing, it causes
dissipation of energy from the electromagnetic wave into the medium. The combined
effect of scattering and absorption is to reduce the amount of energy contained in the
incident wave. If the absolute temperature of the particle differs from zero, electron
transitions from a higher to a lower energy level cause thermal emission of electromagnetic energy at specific frequencies. For complicated systems of molecules with a
large number of degrees of freedom, many different transitions produce spectral emission lines so closely spaced that the resulting radiation spectrum becomes effectively
continuous and includes emitted energy at all frequencies.
Electromagnetic scattering is a complex phenomenon because the secondary
waves generated by each oscillating charge also stimulate oscillations of all other
charges forming the particle. Furthermore, computation of the total scattered field by
superposing the secondary waves must take account of their phase differences, which
change every time the incidence and/or scattering direction is changed. Therefore, the
total scattered radiation depends on the way the charges are arranged to form the particle with respect to the incident and scattered directions.
Since the number of elementary charges forming a micrometer-sized particle is
extremely large, solving the scattering problem directly by computing and superposing all secondary waves is impracticable even with the aid of modern computers.
Fortunately, however, the same problem can be solved using the concepts of macroscopic electromagnetics, which treat the large collection of charges as a macroscopic
body with a specific distribution of the refractive index. In this case the scattered field
3
4
Scattering, Absorption, and Emission of Light by Small Particles
In (a), (b), and (c) a parallel beam of light is incident from the left. (a) Far-field electromagnetic
scattering by an individual particle in the form of a single body or a fixed cluster. (b) Far-field
scattering by a small volume element composed of randomly positioned, widely separated particles. (c) Multiple scattering by a layer of randomly and sparsely distributed particles. On the
left of the layer, diffuse reflected light; on the right of the layer, diffuse transmitted light. On
the far right, the attenuated incident beam. (d) Each individual particle in the layer receives and
scatters both light from the incident beam, somewhat attenuated, and light diffusely scattered
from the other particles.
Foreword to Part I
5
can be computed by solving the Maxwell equations for the macroscopic electromagnetic field subject to appropriate boundary conditions. This approach appears to be
quite manageable and forms the basis of the modern theory of electromagnetic scattering by small particles.
We do not aim to cover all aspects of electromagnetic scattering and absorption by
a small particle and limit our treatment by imposing several well-defined restrictions,
as follows.
●
●
●
●
●
We consider only the scattering of time-harmonic, monochromatic or quasimonochromatic light in that we assume that the amplitude of the incident
electric field is either constant or fluctuates with time much more slowly than
the time factor exp(−iω t ), where ω is the angular frequency and t is time.
It is assumed that electromagnetic scattering occurs without frequency redistribution, i.e., the scattered light has the same frequency as the incident light.
This restriction excludes inelastic scattering phenomena such as Raman and
Brillouin scattering and fluorescence.
We consider only finite scattering particles and exclude such peculiar twodimensional scatterers as infinite cylinders.
It is assumed that the unbounded host medium surrounding the scatterer is
homogeneous, linear, isotropic, and nonabsorbing.
We study only scattering in the far-field zone, where the propagation of the
scattered wave is away from the particle, the electric field vector vibrates in
the plane perpendicular to the propagation direction, and the scattered field
amplitude decays inversely with distance from the particle.
By directly solving the Maxwell equations, one can find the field scattered by an
object in the form of a single body or a fixed cluster consisting of a limited number of
components. However, one often encounters situations in which light is scattered by a
very large group of particles forming a constantly varying spatial configuration. A
typical example is a cloud of water droplets or ice crystals in which the particles are
constantly moving, spinning, and even changing their shapes and sizes due to oscillations of the droplet surface, evaporation, condensation, sublimation, and melting.
Although such a particle collection can be treated at each given moment as a fixed
cluster, a typical measurement of light scattering takes a finite amount of time, over
which the spatial configuration of the component particles and their sizes, orientations, and/or shapes continuously and randomly change. Therefore, the registered
signal is in effect a statistical average over a large number of different cluster realizations.
A logical way of modeling the measurement of light scattering by a random collection of particles would be to solve the Maxwell equations for a statistically representative range of fixed clusters and then take the average. However, this approach
becomes prohibitively time consuming if the number of cluster components is large
and is impractical for objects such as clouds in planetary atmospheres, oceanic hydrosols, or stellar dust envelopes. Moreover, in the traditional far-field scattering formalism a cluster is treated as a single scatterer and it is assumed that the distance from the
6
Scattering, Absorption, and Emission of Light by Small Particles
cluster to the observation point is much larger than any linear dimension of the cluster. This assumption may well be violated in laboratory and remote sensing measurements, thereby making necessary explicit computations of the scattered light in the
near-field zone of the cluster as a whole.
Fortunately, particles forming a random group can often be considered as independent scatterers. This means that the electromagnetic response of each particle in
the group can be calculated using the extinction and phase matrices that describe the
scattering of a plane electromagnetic wave by the same particle but placed in an infinite homogeneous space in complete isolation from all other particles (panel (a) of the
diagram). In general, this becomes possible when (i) each particle resides in the farfield zones of all the other particles forming the group, and (ii) scattering by individual particles is incoherent, i.e., there are no systematic phase relations between partial
waves scattered by individual particles during the time interval necessary to take the
measurement. As a consequence of condition (ii), the intensities (or, more generally,
the Stokes parameters) of the partial waves can be added without regard to phase. An
important exception is scattering in the exact forward direction, which is always coherent and causes attenuation of the incident wave.
The assumption of independent scattering greatly simplifies the problem of computing light scattering by groups of randomly positioned, widely separated particles.
Consider first the situation when a plane wave illuminates a small volume element
containing a tenuous particle collection, as depicted schematically in panel (b) of the
diagram. Each particle is excited by the external field and the secondary fields scattered by all other particles. However, if the number of particles is sufficiently small
and their separation is sufficiently large then the contribution of the secondary waves
to the field exciting each particle is much smaller than the external field. Therefore,
the total scattered field can be well approximated by the sum of the fields generated
by the individual particles in response to the external field in isolation from the other
particles. This approach is called the single-scattering approximation. By assuming
also that particle positions are sufficiently random, one can show that the optical cross
sections and the extinction and phase matrices of the volume element are obtained by
simply summing the respective characteristics of all constituent particles.
When the scattering medium contains very many particles, the single-scattering
approximation is no longer valid. Now one must explicitly take into account that each
particle is illuminated by light scattered by other particles as well as by the (attenuated) incident light, as illustrated in panels (c) and (d) of the diagram. This means that
each particle scatters light that has already been scattered by other particles, so that
the light inside the scattering medium and the light leaving the medium have a significant multiply scattered (or diffuse) component. A traditional approach in this case
is to find the intensity and other Stokes parameters of the diffuse light by solving the
so-called radiative transfer equation. This technique still assumes that particles forming the scattering medium are randomly positioned and widely separated and that the
extinction and phase matrices of each small volume element can be obtained by incoherently adding the respective characteristics of the constituent particles.
Thus, the treatment of light scattering by a large group of randomly positioned,
Foreword to Part I
7
widely separated particles can be partitioned into three consecutive steps:
●
●
●
computation of the far-field scattering and absorption properties of an individual particle by solving the Maxwell equations;
computation of the scattering and absorption properties of a small volume
element containing a tenuous particle collection by using the single-scattering
approximation; and
computation of multiple scattering by the entire particle group by solving the
radiative transfer equation supplemented by appropriate boundary conditions.
Although the last two steps are inherently approximate, they are far more practicable
than attempting to solve the Maxwell equations for large particle collections and usually provide results accurate enough for many applications. A notable exception is the
exact backscattering direction, where so-called self-avoiding reciprocal multiplescattering paths in the particle collection always interfere constructively and cause a
coherent intensity peak. This phenomenon is called coherent backscattering (or weak
photon localization) and is not explicitly described by the standard radiative transfer
theory.
When a group of randomly moving and spinning particles is illuminated by a
monochromatic, spatially coherent plane wave (e.g., laser light), the random constructive and destructive interference of the light scattered by individual particles
generates in the far-field zone a speckle pattern that fluctuates in time and space. In
this book we eliminate the effect of fluctuations by assuming that the Stokes parameters of the scattered light are averaged over a period of time much longer than the
typical period of the fluctuations. In other words, we deal with the average, static
component of the scattering pattern. Therefore, the subject of the book could be called
static light scattering. Although explicit measurements of the spatial and temporal
fluctuations of the speckle pattern are more complicated than measurements of the
averages, they can contain useful information about the particles complementary to
that carried by the mean Stokes parameters. Statistical analyses of light scattered by
systems of suspended particles are the subject of the discipline called photon correlation spectroscopy (or dynamic light scattering) and form the basis of many well established experimental techniques. For example, instruments for the measurement of
particle size and dispersity and laser Doppler velocimeters and transit anemometers
have been commercially available for many years. More recent research has demonstrated the application of polarization fluctuation measurements to particle shape
characterization (Pitter et al. 1999; Jakeman 2000). Photon correlation spectroscopy is
not discussed in this volume; the interested reader can find the necessary information
in the books by Cummins and Pike (1974, 1977), Pecora (1985), Brown (1993), Pike
and Abbiss (1997), and Berne and Pecora (2000) as well as in the recent feature issues
of Applied Optics edited by Meyer et al. (1997, 2001).
Chapter 1
Polarization characteristics of electromagnetic
radiation
The analytical and numerical basis for describing scattering properties of media composed of small discrete particles is formed by the classical electromagnetic theory.
Although there are several excellent textbooks outlining the fundamentals of this theory, it is convenient for our purposes to begin with a summary of those concepts and
equations that are central to the subject of this book and will be used extensively in
the following chapters.
We start by formulating Maxwell’s equations and constitutive relations for timeharmonic macroscopic electromagnetic fields and derive the simplest plane-wave
solution, which underlies the basic optical idea of a monochromatic parallel beam of
light. This solution naturally leads to the introduction of such fundamental quantities
as the refractive index and the Stokes parameters. Finally, we define the concept of a
quasi-monochromatic beam of light and discuss its implications.
1.1
Maxwell’s equations, time-harmonic fields, and the
Poynting vector
The mathematical description of all classical optics phenomena is based on the set of
Maxwell’s equations for the macroscopic electromagnetic field at interior points in
matter, which in SI units are as follows (Jackson 1998):
∇ ⋅ D = ρ,
∇×E = −
∇ ⋅ B = 0,
8
(1.1)
∂B
,
∂t
(1.2)
(1.3)
1 Polarization characteristics of electromagnetic radiation
∇×H = J +
∂D
,
∂t
9
(1.4)
where t is time, E the electric and H the magnetic field, B the magnetic induction, D
the electric displacement, and ρ and J the macroscopic (free) charge density and
current density, respectively. All quantities entering Eqs. (1.1)–(1.4) are functions of
time and spatial coordinates. Implicit in the Maxwell equations is the continuity
equation
∂ρ
+ ∇ ⋅ J = 0,
∂t
(1.5)
which can be derived by combining the time derivative of Eq. (1.1) with the divergence of Eq. (1.4). The vector fields entering Eqs. (1.1)–(1.4) are related by
D = ε 0 E + P,
H=
1
B − M,
µ0
(1.6)
(1.7)
where P is the electric polarization (average electric dipole moment per unit volume),
M is the magnetization (average magnetic dipole moment per unit volume), and ε 0
and µ 0 are the electric permittivity and the magnetic permeability of free space.
Equations (1.1)–(1.7) are insufficient for a unique determination of the electric and
magnetic fields from a given distribution of charges and currents and must be supplemented with so-called constitutive relations:
J = σ E,
(1.8)
B = µH ,
(1.9)
P = ε 0 χ E,
(1.10)
where σ is the conductivity, µ the permeability, and χ the electric susceptibility.
For linear and isotropic media, σ , µ , and χ are scalars independent of the fields.
The microphysical derivation and the range of validity of the macroscopic Maxwell
equations are discussed in detail by Jackson (1998).
The Maxwell equations are strictly valid only for points in whose neighborhood
the physical properties of the medium, as characterized by σ , µ , and χ , vary continuously. Across an interface separating one medium from another the field vectors
E, D, B, and H may be discontinuous. The boundary conditions at such an interface
can be derived from the integral equivalents of the Maxwell equations (Jackson 1998)
and are as follows:
1. There is a discontinuity in the normal component of D:
(D 2 − D1) ⋅ nˆ = ρ S ,
(1.11)
Scattering, Absorption, and Emission of Light by Small Particles
10
where n̂ is the unit vector directed along the local normal to the interface separating media 1 and 2 and pointing toward medium 2 and ρ S is the surface
charge density (the charge per unit area).
2. There is a discontinuity in the tangential component of H:
nˆ × (H 2 − H1) = J S ,
(1.12)
where J S is the surface current density. However, media with finite conductivity cannot support surface currents, so that
nˆ × (H 2 − H1) = 0 (finite conductivity).
(1.13)
3. The normal component of B and the tangential component of E are continuous:
(B 2 − B1) ⋅ nˆ = 0,
(1.14)
nˆ × (E 2 − E1) = 0.
(1.15)
The boundary conditions (1.11)–(1.15) are useful in solving the Maxwell equations in
different adjacent regions with continuous physical properties and then linking the
partial solutions to determine the fields throughout all space.
We assume that all fields and sources are time-harmonic and adopt the standard
practice of representing real time-dependent fields as real parts of the respective complex fields, viz.,
E(r, t ) = Re E c (r, t ) = Re[E(r )e −iω t ] ≡ 12 [E(r )e −iω t + E ∗ (r )e iω t ],
(1.16)
where r is the position (radius) vector, ω the angular frequency, i = − 1, and the
asterisk denotes a complex-conjugate value. Then we can derive from Eqs. (1.1)–
(1.10)
∇ ⋅ D(r ) = ρ (r )
or
∇ ⋅ [ε E(r )] = 0,
(1.17)
∇ × E(r ) = iωµH (r ),
(1.18)
∇ ⋅ [ µH (r )] = 0,
(1.19)
∇ × H (r ) = J (r ) − iω D(r ) = −iωε E(r ),
(1.20)
where
ε = ε 0 (1 + χ ) + i
σ
ω
(1.21)
is the (complex) permittivity. Under the complex time-harmonic representation, the
constitutive coefficients σ , µ , and χ can be frequency dependent and are not restricted to be real (Jackson 1998). For example, a complex permeability implies a
difference in phase between the real time-harmonic magnetic field H and the corresponding real time-harmonic magnetic induction B. We will show later that complex
ε and/or µ results in a non-zero imaginary part of the refractive index, Eq. (1.44),
1 Polarization characteristics of electromagnetic radiation
11
thereby causing the absorption of electromagnetic energy, Eq. (1.45), by converting it
into other forms of energy such as heat.
Note that the scalar or the vector product of two real vector fields is not equal to
the real part of the respective product of the corresponding complex vector fields.
Instead we have
c(r, t ) = a(r, t ) ⋅ b(r, t )
= 14 [a(r )e −iω t + a ∗ (r )e iω t ] ⋅ [b(r )e −iω t + b ∗ (r )e iω t ]
= 12 Re[a(r ) ⋅ b ∗ (r ) + a(r ) ⋅ b(r )e −2iω t ]
(1.22)
and similarly for a vector product. A common situation in practice is that the angular
frequency ω is so high that a measuring instrument is not capable of following the
rapid oscillations of the instantaneous product values but rather responds to a time
average
ác(r )ñ =
1
∆t
t + ∆t
dt ′c(r, t ′ ),
(1.23)
t
where ∆t is a time interval long compared with 1 ω . Therefore, it follows from Eq.
(1.22) that for time averages of products, one must take the real part of the product of
one complex field with the complex conjugate of the other, e.g.,
ác(r )ñ = 12 Re[a(r ) ⋅ b ∗ (r )].
(1.24)
The flow of the electromagnetic energy is described by the so-called Poynting
vector S. The expression for S can be derived by considering the conservation of
energy and taking into account that the magnetic field can do no work and that for a
local charge q the rate of doing work by the electric field is q (r ) v (r ) ⋅ E(r, t ), where v
is the velocity of the charge. Accordingly, consider the integral
1
2
dV J ∗(r ) ⋅ E(r )
(1.25)
V
over a finite volume V, whose real part gives the time-averaged rate of work done by
the electromagnetic field and which must be balanced by the corresponding rate of
decrease of the electromagnetic energy within V. Using Eqs. (1.18) and (1.20) and
the vector identity
∇ ⋅ (a × b) = b ⋅ (∇ × a) − a ⋅ (∇ × b),
(1.26)
we derive
1
2
dV J ∗(r ) ⋅ E(r ) =
V
1
2
1
=
2
dV E(r ) ⋅ [∇ × H ∗(r ) − iω D∗(r )]
V
dV {−∇ ⋅ [E(r ) × H∗(r )] − iω [E(r ) ⋅ D∗(r ) − B(r ) ⋅ H∗(r )]}.
V
(1.27)
12
Scattering, Absorption, and Emission of Light by Small Particles
If we now define the complex Poynting vector
S(r ) = 12 [E(r ) × H ∗(r )]
(1.28)
and the harmonic electric and magnetic energy densities
we (r ) = 14 [E(r ) ⋅ D∗(r )],
wm (r ) = 14 [B(r ) ⋅ H ∗(r )],
(1.29)
and use the Gauss theorem, we have instead of Eq. (1.27)
1
2
dV J ∗(r ) ⋅ E(r ) +
V
dS S(r ) ⋅ nˆ + 2iω
S
dV [ we (r ) − wm (r )] = 0,
(1.30)
V
where the closed surface S encloses the volume V and n̂ is a unit vector in the direction of the local outward normal to the surface. The real part of Eq. (1.30) manifests
the conservation of energy for the time-averaged quantities by requiring that the rate
of the total work done by the fields on the sources within the volume, the electromagnetic energy flowing out through the volume boundary per unit time, and the time rate
of change of the electromagnetic energy within the volume add up to zero. The timeaveraged Poynting vector áS(r )ñ is equal to the real part of the complex Poynting
vector,
áS(r )ñ = Re[S(r )],
and has the dimension of [energy/(area × time)]. The net rate W at which the electromagnetic energy crosses the surface S is
W =−
dS áS(r )ñ ⋅ nˆ .
(1.31)
S
The rate is positive if there is a net transfer of electromagnetic energy into the volume
V and is negative otherwise.
1.2
Plane-wave solution
A fundamental feature of the Maxwell equations is that they allow for a simple traveling-wave solution, which represents the transport of electromagnetic energy from
one point to another and embodies the concept of a perfectly monochromatic parallel
beam of light. This solution is a plane electromagnetic wave propagating in a homogeneous medium without sources and is given by
E c (r, t ) = E 0 exp(ik ⋅ r − iω t ),
H c (r, t ) = H 0 exp(ik ⋅ r − iω t ),
(1.32)
where E0 and H 0 are constant complex vectors. The wave vector k is also constant
and may, in general, be complex:
k = k R + ik I ,
(1.33)
1 Polarization characteristics of electromagnetic radiation
Plane surface normal to K :
r1 ⋅ K = r2 ⋅ K = r3 ⋅ K
13
K
r1
r2
r3
O
Figure 1.1. Plane surface normal to a real vector K.
where k R and k I are real vectors. We thus have
E c (r, t ) = E 0 exp(−k I ⋅ r ) exp(ik R ⋅ r − iω t ),
H c (r, t ) = H 0 exp(−k I ⋅ r ) exp(ik R ⋅ r − iω t ).
(1.34)
(1.35)
E0 exp(−k I ⋅ r ) and H 0 exp(−k I ⋅ r ) are the amplitudes of the electric and magnetic
waves, respectively, while k R ⋅ r − ω t is their phase. Obviously, k R is normal to the
surfaces of constant phase, whereas k I is normal to the surfaces of constant amplitude. (A plane surface normal to a real vector K is defined as r ⋅ K = constant, where
r is the radius vector drawn from the origin of the reference frame to any point in the
plane; see Fig. 1.1.) Surfaces of constant phase propagate in the direction of k R with
the phase velocity v = ω |k R |. The electromagnetic wave is called homogeneous
when k R and k I are parallel (including the case k I = 0); otherwise it is called inhomogeneous. When k R k I , the complex wave vector can be written as k =
(k R + ik I )nˆ , where n̂ is a real unit vector in the direction of propagation and both k R
and k I are real and non-negative.
The Maxwell equations for the plane wave take the form
k ⋅ E 0 = 0,
k ⋅ H 0 = 0,
k × E 0 = ωµH 0,
k × H 0 = −ωε E 0 .
(1.36)
(1.37)
(1.38)
(1.39)
The first two equations indicate that the plane electromagnetic wave is transverse:
both E 0 and H 0 are perpendicular to k. Furthermore, it is evident from Eq. (1.38)
or (1.39) that E 0 and H 0 are mutually perpendicular: E 0 ⋅ H 0 = 0. Since E 0, H 0,
and k are, in general, complex vectors, the physical interpretation of these facts can
be far from obvious. It becomes most transparent when ε , µ , and k are real. The
14
Scattering, Absorption, and Emission of Light by Small Particles
reader can verify that in this case the real field vectors E and H are mutually perpendicular and lie in a plane normal to the direction of wave propagation.
Equations (1.32) and (1.38) yield H c (r, t ) = (ωµ ) −1 k × E c (r, t ). Therefore, a
plane electromagnetic wave can always be considered in terms of only the electric (or
only the magnetic) field.
By taking the vector product of both sides of Eq. (1.38) with k and using Eq.
(1.39) and the vector identity
a × (b × c) = b(a ⋅ c) − c(a ⋅ b),
(1.40)
together with Eq. (1.36), we derive
k ⋅ k = ω 2 εµ .
(1.41)
In the practically important case of a homogeneous plane wave, we obtain from Eq.
(1.41)
k = k R + ik I = ω εµ =
ωm
,
c
(1.42)
where k is the wave number,
c=
1
(1.43)
ε 0µ0
is the speed of light in a vacuum, and
m = mR + imI =
εµ
= c εµ
ε 0µ0
(1.44)
is the complex refractive index with non-negative real part mR and non-negative
imaginary part mI. Thus, the plane homogeneous wave has the form
æ ω
ö
æ ω
ö
E c (r, t ) = E 0 exp ç − mI nˆ ⋅ r ÷ exp ç i mR nˆ ⋅ r − iω t ÷.
è c
ø
è c
ø
(1.45)
If the imaginary part of the refractive index is non-zero, then it determines the decay
of the amplitude of the wave as it propagates through the medium, which is thus absorbing. The real part of the refractive index determines the phase velocity of the
wave: v = c mR . For a vacuum, m = mR = 1 and v = c.
As follows from Eqs. (1.28), (1.32), (1.38), and (1.40), the time-averaged
Poynting vector of a plane wave is
áS(r )ñ = 12 Re[E(r ) × H ∗(r )]
ì k ∗ [E(r ) ⋅ E∗(r )] − E∗(r )[k ∗ ⋅ E(r )] ü
= Reí
ý.
2ωµ ∗
î
þ
If the wave is homogeneous, then k ⋅ E = 0 and so k ∗ ⋅ E = 0, and
(1.46)
1 Polarization characteristics of electromagnetic radiation
ìï ε üï
æ ω
ö
2
áS(r )ñ = 12 Reí
ý |E 0 | exp ç − 2 mI nˆ ⋅ r ÷nˆ .
µ
c
ïî
ïþ
è
ø
15
(1.47)
Thus, áS(r )ñ is in the direction of propagation and its absolute value I (r ) = |áS(r )ñ|,
usually called the intensity (or irradiance), is exponentially attenuated provided that
the medium is absorbing:
I (r ) = I 0 e −α nˆ ⋅r ,
(1.48)
where I 0 is the intensity at r = 0. The absorption coefficient α is
α =2
4π mI
ω
,
mI =
λ
c
(1.49)
where
λ=
2π c
ω
(1.50)
is the free-space wavelength. The intensity has the dimension of monochromatic energy flux: [energy/(area × time)].
The reader can verify that the choice of the time dependence exp(iω t ) rather than
exp(−iω t ) in the complex representation of time-harmonic fields in Eq. (1.16) would
have led to m = mR − imI with a non-negative mI. The exp(−iω t ) time-factor convention adopted here has been used in many other books on optics and light scattering
(e.g., Born and Wolf 1999; Bohren and Huffman 1983; Barber and Hill 1990) and is a
nearly standard choice in electromagnetics (e.g., Stratton 1941; Tsang et al. 1985;
Kong 1990; Jackson 1998) and solid-state physics. However, van de Hulst (1957)
and Kerker (1969) used the time factor exp(iω t ), which implies a non-positive
imaginary part of the complex refractive index. It does not matter in the final analysis
which convention is chosen because all measurable quantities of practical interest are
always real. However, it is important to remember that once a choice of the time
factor has been made, its consistent use throughout all derivations is essential.
1.3
Coherency matrix and Stokes parameters
Most photometric and polarimetric optical instruments cannot measure the electric
and magnetic fields associated with a beam of light; rather, they measure quantities
that are time averages of real-valued linear combinations of products of field vector
components and have the dimension of intensity. Important examples of such observable quantities are so-called Stokes parameters. In order to define them, we will use
the spherical coordinate system associated with a local right-handed Cartesian coordinate system having its origin at the observation point, as shown in Fig. 1.2. The direction of propagation of a plane electromagnetic wave in a homogeneous nonab-
16
Scattering, Absorption, and Emission of Light by Small Particles
z
ϕ̂
ϑ
n̂ = ϑ̂ × ϕ̂
O
ϑ̂
y
ϕ
x
Figure 1.2. Coordinate system used to describe the direction of propagation and the polarization state of a plane electromagnetic wave.
sorbing medium is specified by a unit vector n̂ or, equivalently, by a couplet (ϑ , ϕ ),
where ϑ ∈ [0, π ] is the polar (zenith) angle measured from the positive z-axis and
ϕ ∈ [0, 2π ) is the azimuth angle measured from the positive x-axis in the clockwise
direction when looking in the direction of the positive z-axis. Since the medium is
assumed to be nonabsorbing, the component of the electric field vector along the direction of propagation n̂ is equal to zero, so that the electric field at the observation
point is given by E = Eϑ + Eϕ , where Eϑ and Eϕ are the ϑ - and ϕ - components of
the electric field vector. The component Eϑ = Eϑ ϑ̂ lies in the meridional plane (i.e.,
plane through n̂ and the z-axis), whereas the component Eϕ = Eϕ ϕ̂ is perpendicular
to this plane; ϑ̂ and ϕ̂ are the corresponding unit vectors such that nˆ = ϑˆ × ϕˆ . Note
that in the microwave remote sensing literature, Eϑ and Eϕ are often denoted as E v
and E h and called the vertical and horizontal electric field vector components, respectively (e.g., Tsang et al. 1985; Ulaby and Elachi 1990).
The specification of a unit vector n̂ uniquely determines the meridional plane of
the propagation direction except when n̂ is oriented along the positive or negative
direction of the z-axis. Although it may seem redundant to specify ϕ in addition to ϑ
when ϑ = 0 or π , the unit ϑ and ϕ vectors and, thus, the electric field vector components Eϑ and Eϕ still depend on the orientation of the meridional plane. Therefore, we will always assume that the specification of n̂ implicitly includes the specification of the appropriate meridional plane in cases when n̂ is parallel to the z-axis.
To minimize confusion, we often will specify explicitly the direction of propagation
using the angles ϑ and ϕ ; the latter uniquely defines the meridional plane when
ϑ = 0 or π .
Consider a plane electromagnetic wave propagating in a medium with constant
real ε , µ , and k and given by
1 Polarization characteristics of electromagnetic radiation
E c (r, t ) = E 0 exp(iknˆ ⋅ r − iω t ).
17
(1.51)
The simplest complete set of linearly independent quadratic combinations of the
electric field vector components with non-zero time averages consists of the following four quantities:
Ecϑ Ec∗ϑ = E0ϑ E0∗ϑ , Ecϑ Ec∗ϕ = E0ϑ E0∗ϕ , Ecϕ Ec∗ϑ = E0ϕ E0∗ϑ , Ecϕ Ec∗ϕ = E0ϕ E0∗ϕ .
ε µ have the dimension of monochromatic
energy flux and form the 2 × 2 so-called coherency (or density) matrix ρ:
The products of these quantities and
é ρ11
ρ=ê
ë ρ 21
1
2
ρ12 ù 1 ε é E0ϑ E0∗ϑ
ê
ú=
ρ 22 û 2 µ êë E0ϕ E0∗ϑ
E0ϑ E0∗ϕ ù
ú.
E0ϕ E0∗ϕ ûú
(1.52)
The completeness of the set of the four coherency matrix elements means that any
plane-wave characteristic directly observable with a traditional optical instrument is a
real-valued linear combination of these quantities.
Since ρ12 and ρ 21 are, in general, complex, it is convenient to introduce an alternative complete set of four real, linearly independent quantities called Stokes parameters. Let us first group the elements of the 2 × 2 coherency matrix into a 4 × 1
coherency vector (O’Neill 1992):
é ρ11 ù
ê ú
ρ12
1
J=ê ú=
ê ρ 21 ú 2
ê ú
ëê ρ 22 ûú
é E0ϑ E0∗ϑ ù
ê
∗ ú
ε ê E0ϑ E0ϕ ú
.
µ ê E0ϕ E0∗ϑ ú
ú
ê
êë E0ϕ E0∗ϕ úû
(1.53)
The Stokes parameters I, Q, U, and V are then defined as the elements of a 4× 1 column vector I, otherwise known as the Stokes vector, as follows:
éI ù
ê ú
Q
1
I = ê ú = DJ =
êU ú
2
ê ú
ëêV ûú
é E0ϑ E0∗ϑ + E0ϕ E0∗ϕ ù
ê
∗
∗ ú
ε ê E0ϑ E0ϑ − E0ϕ E0ϕ ú 1
=
µ ê − E0ϑ E0∗ϕ − E0ϕ E0∗ϑ ú 2
ú
ê
êëi( E0ϕ E0∗ϑ − E0ϑ E0∗ϕ )úû
é E0ϑ E0∗ϑ + E0ϕ E0∗ϕ ù
ê
∗
∗ ú
ε ê E0ϑ E0ϑ − E0ϕ E0ϕ ú
,
µ ê − 2 Re( E0ϑ E0∗ϕ ) ú
ú
ê
êë 2Im(E0ϑ E0∗ϕ ) úû
(1.54)
where
0 1ù
é1 0
ê
ú
1 0
0 − 1ú
ê
D=
.
ê0 − 1 − 1 0 ú
ê
ú
0 úû
êë0 − i i
(1.55)
The converse relationship is
J = D −1I,
(1.56)
18
Scattering, Absorption, and Emission of Light by Small Particles
where the inverse matrix D −1 is given by
0 0ù
é1 1
ê
ú
1 ê0 0 − 1 i ú
−1
D =
.
2 ê0 0 − 1 − i ú
ê
ú
êë1 − 1 0 0 úû
(1.57)
Since the Stokes parameters are real-valued and have the dimension of monochromatic energy flux, they can be measured directly with suitable optical instruments. Furthermore, they form a complete set of quantities needed to characterize a
plane electromagnetic wave, inasmuch as it is subject to practical analysis. This
means that (i) any other observable quantity is a linear combination of the four Stokes
parameters, and (ii) it is impossible to distinguish between two plane waves with the
same values of the Stokes parameters using a traditional optical device (the so-called
principle of optical equivalence). Indeed, the two complex amplitudes
E0ϑ = aϑ exp(i∆ ϑ ) and E0ϕ = aϕ exp(i∆ ϕ ) are characterized by four real numbers:
the non-negative amplitudes aϑ and aϕ and the phases ∆ϑ and ∆ ϕ = ∆ ϑ − ∆. The
Stokes parameters carry information about the amplitudes and the phase difference
∆, but not about ∆ϑ . The latter is the only quantity that could be used to distinguish
different waves with the same aϑ , aϕ , and ∆ (and thus the same Stokes parameters),
but it vanishes when a field vector component is multiplied by the complex conjugate
value of the same or another field vector component; cf. Eqs. (1.52) and (1.54).
The first Stokes parameter, I, is the intensity introduced in the previous section;
the explicit definition given in Eq. (1.54) is applicable to a homogeneous, nonabsorbing medium. The Stokes parameters Q, U, and V describe the polarization state
of the wave. The ellipsometric interpretation of the Stokes parameters will be the
subject of the following section. The reader can easily verify that the Stokes parameters of a plane monochromatic wave are not completely independent but rather
are related by the quadratic identity
I 2 ≡ Q 2 + U 2 + V 2.
(1.58)
We will see later, however, that this identity may not hold for a quasi-monochromatic
beam of light. Because one usually must deal with relative rather than absolute intensities, the constant factor
1
2
ε µ is often unimportant and will be omitted in all
cases where this does not generate confusion.
The coherency matrix and the Stokes vector are not the only representations of
polarization and not always the most convenient ones. Two other frequently used
representations are the real so-called modified Stokes column vector given by
IMS
é 12 ( I + Q)ù
éI v ù
ê1
ú
ê ú
Ih ú
( I − Q)ú
ê
ê
2
=
= BI = ê
ú
êU ú
U
ê
ú
ê ú
êë V
úû
êë V úû
(1.59)
1 Polarization characteristics of electromagnetic radiation
19
and the complex circular-polarization column vector defined as
ICP
é I2 ù
ê ú
I0
1
= ê ú = AI =
ê I −0 ú
2
ê ú
ëê I −2 ûú
éQ + iU ù
ê
ú
ê I + V ú,
ê I −V ú
ê
ú
ëêQ − iU ûú
(1.60)
where
é1 2
ê
12
B=ê
ê0
ê
êë 0
é0
ê
1 ê1
A=
2 ê1
ê
ëê0
0 0ù
ú
− 1 2 0 0ú
,
0
1 0ú
ú
0
0 1úû
1 i
0ù
ú
0 0 1ú
.
0 0 − 1ú
ú
1 − i 0 ûú
12
(1.61)
(1.62)
It is easy to verify that
I = B −1IMS
(1.63)
I = A −1ICP ,
(1.64)
and
where
é1 1
ê
1 −1
−1
B =ê
ê0 0
ê
êë0 0
0 0ù
ú
0 0ú
1 0ú
ú
0 1úû
(1.65)
0ù
ú
0 0 1ú
.
0 0 iú
ú
1 − 1 0úû
(1.66)
and
A −1
1.4
é0
ê
1
=ê
ê− i
ê
êë 0
1
1
Ellipsometric interpretation of Stokes parameters
In this section we show how the Stokes parameters can be used to derive the ellipsometric characteristics of the plane electromagnetic wave given by Eq. (1.51).
20
Scattering, Absorption, and Emission of Light by Small Particles
Writing
E0ϑ = aϑ exp(i∆ ϑ ),
(1.67)
E0ϕ = aϕ exp(i∆ ϕ )
(1.68)
with real non-negative amplitudes aϑ and aϕ and real phases ∆ϑ and ∆ ϕ , using Eq.
(1.54), and omitting the factor
1
2
ε µ we obtain for the Stokes parameters
I = aϑ2 + aϕ2 ,
(1.69)
aϑ2
(1.70)
Q=
−
aϕ2 ,
U = −2aϑ aϕ cos ∆,
(1.71)
V = 2aϑ aϕ sin ∆,
(1.72)
∆ = ∆ϑ − ∆ϕ .
(1.73)
where
Substituting Eqs. (1.67) and (1.68) in Eq. (1.51), we have for the real electric
vector
Eϑ (r, t ) = aϑ cos(δ ϑ − ω t ),
(1.74)
Eϕ (r, t ) = aϕ cos(δ ϕ − ω t ),
(1.75)
δ ϑ = ∆ ϑ + k nˆ ⋅ r,
(1.76)
where
δ ϕ = ∆ ϕ + k nˆ ⋅ r.
At any fixed point O in space, the endpoint of the real electric vector given by Eqs.
(1.74)–(1.76) describes an ellipse with specific major and minor axes and orientation
(see the top panel of Fig. 1.3). The major axis of the ellipse makes an angle ζ with
the positive direction of the ϕ - axis such that ζ ∈ [0, π ). By definition, this orientation angle is obtained by rotating the ϕ - axis in the clockwise direction when looking
in the direction of propagation, until it is directed along the major axis of the ellipse.
The ellipticity is defined as the ratio of the minor to the major axes of the ellipse and
is usually expressed as |tan β |, where β ∈ [− π 4 , π 4]. By definition, β is positive
when the real electric vector at O rotates clockwise, as viewed by an observer looking
in the direction of propagation. The polarization for positive β is called righthanded, as opposed to the left-handed polarization corresponding to the anticlockwise rotation of the electric vector.
To express the orientation ζ of the ellipse and the ellipticity |tan β | in terms of
the Stokes parameters, we first write the equations representing the rotation of the real
electric vector at O in the form
E q (r, t ) = a sinβ sin(δ − ω t ),
(1.77)
E p (r, t ) = a cosβ cos(δ − ω t ),
(1.78)
1 Polarization characteristics of electromagnetic radiation
21
(a) Polarization ellipse
β
ζ
ϕ
p
q
ϑ
(b) Elliptical polarization (V ≠ 0)
Q < 0 U = 0 V< 0
Q>0 U=0 V>0
Q = 0 U > 0 V< 0 Q = 0 U< 0 V > 0
(c) Linear polarization (V = 0)
Q = –I U = 0
Q=I U=0
Q=0 U=I
Q = 0 U = –I
(d) Circular polarization (Q = U = 0)
V = –I
V=I
Figure 1.3. Ellipse described by the tip of the real electric vector at a fixed point O in space
(upper panel) and particular cases of elliptical, linear, and circular polarization. The plane
electromagnetic wave propagates in the direction ϑˆ × ϕˆ (i.e., towards the reader).
where E p and E q are the electric field components along the major and minor axes
of the ellipse, respectively (Fig. 1.3). One easily verifies that a positive (negative) β
22
Scattering, Absorption, and Emission of Light by Small Particles
indeed corresponds to the right-handed (left-handed) polarization. The connection
between Eqs. (1.74)–(1.75) and Eqs. (1.77)–(1.78) can be established by using the
simple transformation rule for rotation of a two-dimensional coordinate system:
Eϑ (r, t ) = − E q (r, t ) cosζ + E p (r, t ) sinζ ,
(1.79)
Eϕ (r, t ) = − E q (r, t ) sinζ − E p (r, t ) cosζ .
(1.80)
By equating the coefficients of cos ω t and sin ω t in the expanded Eqs. (1.74) and
(1.79) and those in the expanded Eqs. (1.75) and (1.80), we obtain
aϑ cos δ ϑ = −a sinβ sinδ cosζ + a cosβ cosδ sinζ ,
(1.81)
aϑ sin δ ϑ = a sinβ cosδ cosζ + a cosβ sinδ sinζ ,
(1.82)
aϕ cos δ ϕ = −a sinβ sinδ sinζ − a cosβ cosδ cosζ ,
(1.83)
aϕ sin δ ϕ = a sinβ cosδ sinζ − a cosβ sinδ cosζ .
(1.84)
Squaring and adding Eqs. (1.81) and (1.82) and Eqs. (1.83) and (1.84) gives
aϑ2 = a 2 (sin 2 β cos 2ζ + cos 2 β sin 2ζ ),
(1.85)
aϕ2 = a 2 (sin 2 β sin 2ζ + cos 2 β cos 2ζ ).
(1.86)
Multiplying Eqs. (1.81) and (1.83) and Eqs. (1.82) and (1.84) and adding yields
aϑ aϕ cos ∆ = − 12 a 2 cos 2 β sin 2ζ .
(1.87)
Similarly, multiplying Eqs. (1.82) and (1.83) and Eqs. (1.81) and (1.84) and subtracting gives
aϑ aϕ sin ∆ = − 12 a 2 sin 2 β .
(1.88)
Comparing Eqs. (1.69)–(1.72) with Eqs. (1.85)–(1.88), we finally derive
I = a 2,
(1.89)
Q = − I cos 2 β cos 2ζ ,
(1.90)
U = I cos 2 β sin 2ζ ,
(1.91)
V = − I sin 2 β .
(1.92)
The parameters of the polarization ellipse are thus expressed in terms of the
Stokes parameters as follows. The major and minor axes are given by
I cos β and
I |sin β |, respectively (cf. Eqs. (1.77) and (1.78)). Equations (1.90) and (1.91) yield
tan 2ζ = −
U
.
Q
(1.93)
Because | β | ≤ π 4, we have cos 2 β ≥ 0 so that cos 2ζ has the same sign as –Q.
1 Polarization characteristics of electromagnetic radiation
23
Therefore, from the different values of ζ that satisfy Eq. (1.93) but differ by π 2 ,
we must choose the one that makes the sign of cos 2ζ the same as that of –Q. The
ellipticity and handedness follow from
tan 2 β = −
V
Q2 + U 2
.
(1.94)
Thus, the polarization is left-handed if V is positive and right-handed if V is negative
(Fig. 1.3).
The electromagnetic wave becomes linearly polarized when β = 0; then the electric vector vibrates along a line making an angle ζ with the ϕ - axis (cf. Fig. 1.3) and
V = 0. Furthermore, if ζ = 0 or ζ = π 2 then U vanishes as well. This explains the
usefulness of the modified Stokes representation of polarization given by Eq. (1.59) in
situations involving linearly polarized light, as follows. The modified Stokes vector
then has only one non-zero element and is equal to [ I 0 0 0]T if ζ = π 2 (the electric vector vibrates along the ϑ - axis, i.e., in the meridional plane) or to [0 I 0 0]T if
ζ = 0 (the electric vector vibrates along the ϕ - axis, i.e., in the plane perpendicular
to the meridional plane), where T indicates the transpose of a matrix.
If, however, β = ± π 4 , then both Q and U vanish, and the electric vector describes a circle either in the clockwise direction ( β = π 4 , V = − I ) or the anticlockwise direction ( β = −π 4 , V = I ), as viewed by an observer looking in the direction of propagation (Fig. 1.3). In this case the electromagnetic wave is circularly
polarized; the circular-polarization vector ICP has only one non-zero element and
takes the values [0 0 I 0]T and [0 I 0 0]T , respectively (see Eq. (1.60)).
The polarization ellipse, along with a designation of the rotation direction (rightor left-handed), fully describes the temporal evolution of the real electric vector at a
fixed point in space. This evolution can also be visualized by plotting the curve, in
(ϑ , ϕ , t ) coordinates, described by the tip of the electric vector as a function of time.
For example, in the case of an elliptically polarized plane wave with right-handed
polarization the curve is a right-handed helix with an elliptical projection onto the
ϑϕ - plane centered around the t-axis (Fig. 1.4(a)). The pitch of the helix is simply
2π ω , where ω is the angular frequency of the wave. Another way to visualize a
plane wave is to fix a moment in time and draw a three-dimensional curve in
(ϑ , ϕ , s ) coordinates described by the tip of the electric vector as a function of a spatial coordinate s = r ⋅ nˆ oriented along the direction of propagation n̂. According to
Eqs. (1.74)–(1.76), the electric field is the same for all position–time combinations
with constant ks − ω t. Therefore, at any instant of time (say, t = 0) the locus of the
points described by the tip of the electric vector originating at different points on the
s-axis is also a helix, with the same projection onto the ϑϕ - plane as the respective
helix in the (ϑ , ϕ , t ) coordinates but with opposite handedness. For example, for the
wave with right-handed elliptical polarization shown in Fig. 1.4(a), the respective
curve in the (ϑ , ϕ , s ) coordinates is a left-handed elliptical helix, shown in Fig.
24
Scattering, Absorption, and Emission of Light by Small Particles
ϕ
(a)
ϑ
t
ϕ
n̂
(b)
ϑ
s
ζ
n̂
ϕ
(c)
ϑ
s
Figure 1.4. (a) The helix described by the tip of the real electric vector of a plane electromagnetic wave with right-handed polarization in (ϑ , ϕ , t ) coordinates at a fixed point in space. (b)
As in (a), but in (ϑ , ϕ , s ) coordinates at a fixed moment in time. (c) As in (b), but for a linearly polarized wave.
1.4(b). The pitch of this helix is the wavelength λ . It is now clear that the propagation of the wave in time and space can be represented by progressive movement in
time of the helix shown in Fig. 1.4(b) in the direction of n̂ with the speed of light.
With increasing time, the intersection of the helix with any plane s = constant describes a right-handed vibration ellipse. In the case of a circularly polarized wave, the
elliptical helix becomes a helix with a circular projection onto the ϑϕ - plane. If the
wave is linearly polarized, then the helix degenerates into a simple sinusoidal curve in
the plane making an angle ζ with the ϕ - axis (Fig. 1.4(c)).
1.5
Rotation transformation rule for Stokes parameters
The Stokes parameters of a plane electromagnetic wave are always defined with respect to a reference plane containing the direction of wave propagation. If the reference plane is rotated about the direction of propagation then the Stokes parameters are
modified according to a rotation transformation rule, which can be derived as follows.
Consider a rotation of the coordinate axes ϑ and ϕ through an angle 0 ≤ η < 2π in
1 Polarization characteristics of electromagnetic radiation
ϕ̂ ′
ϕ̂
25
n̂
η
O
η
ϑ̂′
ϑ̂
Figure 1.5. Rotation of the ϑ - and ϕ - axes through an angle η ≥ 0 around n̂ in the clockwise direction when looking in the direction of propagation.
the clockwise direction when looking in the direction of propagation (Fig. 1.5). The
transformation rule for rotation of a two-dimensional coordinate system yields
E0′ϑ = E0ϑ cos η + E0ϕ sin η ,
(1.95)
E0′ϕ = − E0ϑ sin η + E0ϕ cos η ,
(1.96)
where the primes denote the electric field vector components with respect to the new
reference frame. It then follows from Eq. (1.54) that the rotation transformation rule
for the Stokes parameters is
0
é I′ ù
é1
ê ú
ê
Q′
0 cos 2η
I′ = ê ú = L(η )I = ê
êU ′ú
ê0 sin 2η
ê ú
ê
0
ëêV ′ ûú
ëê0
0ù é I ù
úê ú
− sin 2η 0ú êQ ú
,
cos 2η 0ú êU ú
úê ú
0
1ûú ëêV ûú
0
(1.97)
where L(η ) is called the Stokes rotation matrix for angle η. It is obvious that a
η = π rotation does not change the Stokes parameters.
Because
(IMS )′ = BI′ = BL(η )I = BL(η )B −1IMS,
(1.98)
the rotation matrix for the modified Stokes vector is given by
écos 2 η
ê 2
sin η
MS
−1
L (η ) = BL(η )B = êê
sin 2η
ê
êë 0
sin 2 η
cos 2 η
− sin 2η
0
− 12 sin 2η 0ù
ú
1
0ú
2 sin 2η
.
cos 2η
0ú
ú
0
1úû
(1.99)
Similarly, for the circular polarization representation,
(ICP )′ = AI′ = AL(η )I = AL(η ) A −1ICP ,
(1.100)
and the corresponding rotation matrix is diagonal (Hovenier and van der Mee 1983):
26
Scattering, Absorption, and Emission of Light by Small Particles
LCP (η ) = AL(η ) A −1
1.6
éexp(i 2η )
ê
0
=ê
ê
0
ê
0
êë
ù
ú
1 0
0
ú.
ú
0 1
0
ú
0 0 exp(−i 2η )úû
0 0
0
(1.101)
Quasi-monochromatic light and incoherent addition
of Stokes parameters
The definition of a monochromatic plane electromagnetic wave given by Eqs. (1.51)
and (1.67)–(1.68) implies that the complex amplitude E 0 and, therefore, the quantities aϑ , aϕ , ∆ϑ , and ∆ ϕ are constant. In reality, these quantities often fluctuate in
time. Although the typical frequency of these fluctuations is much smaller than the
angular frequency ω , it is still so high that most optical devices are incapable of
tracing the instantaneous values of the Stokes parameters but rather measure averages
of the Stokes parameters over a relatively long period of time. Therefore, we must
modify the definition of the Stokes parameters for such quasi-monochromatic beam
of light as follows:
I = á E0ϑ E0∗ϑ ñ + á E0ϕ E0∗ϕ ñ = á aϑ2 ñ + á aϕ2 ñ,
(1.102)
Q = á E0ϑ E0∗ϑ ñ − á E0ϕ E0∗ϕ ñ = á aϑ2 ñ − á aϕ2 ñ,
(1.103)
U = −á E0ϑ E0∗ϕ ñ − á E0ϕ E0∗ϑ ñ = −2á aϑ aϕ cos ∆ñ ,
(1.104)
V = iá E0ϕ E0∗ϑ ñ − iá E0ϑ E0∗ϕ ñ = 2á aϑ aϕ sin ∆ñ ,
(1.105)
where we have omitted the common factor
áfñ=
1
T
t +T
1
2
ε µ and
dt ′ f (t ′ )
(1.106)
t
denotes the average over a time interval T long compared with the typical period of
fluctuation.
The identity (1.58) is not valid, in general, for a quasi-monochromatic beam. Indeed, now we have
I 2 − Q2 − U 2 − V 2
= 4 [á aϑ2 ñá aϕ2 ñ − á aϑ aϕ cos ∆ñ 2 − á aϑ aϕ sin ∆ñ 2 ]
=
4
T2
t +T
t
t +T
dt ′
t
dt ′′{ [aϑ (t ′ )]2 [aϕ (t ′′ )]2
− aϑ (t ′ ) aϕ (t ′ ) cos[ ∆ (t ′ )]aϑ (t ′′ ) aϕ (t ′′ ) cos[ ∆ (t ′′ )]
−aϑ (t ′ ) aϕ (t ′ ) sin[∆ (t ′ )]aϑ (t ′′ ) aϕ (t ′′ ) sin[∆ (t ′′ )]}
1 Polarization characteristics of electromagnetic radiation
=
t +T
4
T2
t +T
dt ′
t
t
27
dt ′′{[ aϑ (t ′ )]2 [aϕ (t ′′ )]2
−aϑ (t ′ ) aϕ (t ′ ) aϑ (t ′′ ) aϕ (t ′′ ) cos[∆ (t ′ ) − ∆ (t ′′ )]}
=
t +T
2
T2
t +T
dt ′
t
t
dt ′′{ [aϑ (t ′ )]2 [aϕ (t ′′ )]2 + [aϑ (t ′′ )]2 [aϕ (t ′ )]2
−2aϑ (t ′ ) aϕ (t ′ ) aϑ (t ′′ ) aϕ (t ′′ ) cos[∆ (t ′ ) − ∆ (t ′′ )]}
≥
t +T
2
T2
t +T
dt ′
t
t
dt ′′{[aϑ (t ′ )]2 [aϕ (t ′′ )]2 + [aϑ (t ′′ )]2 [aϕ (t ′ )]2
−2aϑ (t ′ ) aϕ (t ′ ) aϑ (t ′′ ) aϕ (t ′′ )}
=
t +T
2
T2
t
t +T
dt ′
t
dt ′′ [aϑ (t ′ ) aϕ (t ′′ ) − aϑ (t ′′ ) aϕ (t ′ )]2
≥ 0,
thereby yielding
I 2 ≥ Q 2 + U 2 + V 2.
(1.107)
The equality holds only if the ratio aϑ (t ) aϕ (t ) of the real amplitudes and the phase
difference ∆(t ) are independent of time, which means that E 0ϑ (t ) and E0ϕ (t ) are
completely correlated. In this case the beam is said to be fully (or completely) polarized. This definition includes a monochromatic wave, but is, of course, more general.
However, if aϑ (t ), aϕ (t ), ∆ ϑ (t ), and ∆ ϕ (t ) are totally uncorrelated and á aϑ2 ñ =
á aϕ2 ñ, then Q = U = V = 0, and the quasi-monochromatic beam of light is said to be
unpolarized (or natural). This means that the parameters of the vibration ellipse traced
by the endpoint of the electric vector fluctuate in such a way that there is no preferred
vibration ellipse.
When two or more quasi-monochromatic beams propagating in the same direction
are mixed incoherently (i.e., there is no permanent phase relationship between the
separate beams), the Stokes vector of the mixture is equal to the sum of the Stokes
vectors of the individual beams:
I=
In ,
(1.108)
n
where n numbers the beams. Indeed, inserting Eqs. (1.67) and (1.68) in Eq. (1.54),
we obtain for the total intensity
á aϑ n aϑ m exp[i(∆ ϑ n − ∆ ϑ m )] + aϕ n aϕ m exp[i(∆ ϕ n − ∆ ϕ m )]ñ
I=
n
=
m
á aϑ n aϑ m exp[i(∆ ϑ n − ∆ ϑ m )] + aϕ n aϕ m exp[i(∆ ϕ n − ∆ ϕ m )]ñ. (1.109)
In +
n
n
m≠n
Since the phases of different beams are uncorrelated, the second term on the right-
28
Scattering, Absorption, and Emission of Light by Small Particles
hand side of the relation above vanishes. Hence
I=
I n,
(1.110)
n
and similarly for Q, U, and V. Of course, this additivity rule also applies to the coherency matrix ρ, the modified Stokes vector I MS, and the circular-polarization vector
ICP. An important example demonstrating the application of Eq. (1.108) is the scattering of light by a small volume element containing randomly positioned particles.
The phases of the individual waves scattered by the particles depend on the positions
of the particles. Therefore, if the distribution of the particles is sufficiently random
then the individual scattered waves will be incoherent and the Stokes vectors of the
individual waves will add. The additivity of the Stokes parameters allows us to generalize the principle of optical equivalence (Section 1.3) to quasi-monochromatic light
as follows: it is impossible by means of a traditional optical instrument to distinguish
between various incoherent mixtures of quasi-monochromatic beams that form a
beam with the same Stokes parameters ( I, Q, U , V ). For example, there is only one
kind of unpolarized light, although it can be composed of quasi-monochromatic
beams in an infinite variety of optically indistinguishable ways.
In view of the general inequality (1.107), it is always possible mathematically to
decompose any quasi-monochromatic beam into two parts, one unpolarized, with a
Stokes vector
[ I − Q 2 + U 2 + V 2 0 0 0]T ,
and one fully polarized, with a Stokes vector
[ Q 2 + U 2 + V 2 Q U V ]T .
Thus, the intensity of the fully polarized component is Q 2 + U 2 + V 2 , so that the
degree of (elliptical) polarization of the quasi-monochromatic beam is
P=
Q2 +U 2 +V 2
I
.
(1.111)
We further define the degree of linear polarization as
PL =
Q2 +U 2
I
(1.112)
and the degree of circular polarization as
PC =
V
.
I
(1.113)
P vanishes for unpolarized light and is equal to unity for fully polarized light. For a
partially polarized beam (0 < P < 1) with V ≠ 0, the sign of V indicates the preferen-
1 Polarization characteristics of electromagnetic radiation
29
Stokes parameters : I, Q, U, V
Intensity: I
Degree of
Degree of polarization : P = Q 2 + U 2 + V 2 I
linear polarizati on:
PL = Q 2 + U 2 I
PQ = − Q I (for U = 0)
circular polarizati on: PC = V I
P = 0 : natural light
0 < P < 1: partially polarized light
P = 1: fully polarized light
Preferential ellipticit y : tan 2 β = V
V > 0: left - handed
Preferential handedness:
Q2 + U 2
Preferential orientation of the polarization ellipse:
V = 0: only linear polarizati on
V < 0: right - handed
U > 0 then ζ = π 4
If Q = 0 and
U = 0 then only circular polarizati on
U < 0 then ζ = 3 π 4
If Q ≠ 0 then tan 2ζ = −U Q and sign (cos 2ζ ) = sign (−Q)
Figure 1.6. Analysis of a quasi-monochromatic beam with Stokes parameters I, Q, U, and V.
tial handedness of the vibration ellipses described by the endpoint of the electric vector: a positive V indicates left-handed polarization and a negative V indicates righthanded polarization. By analogy with Eqs. (1.93) and (1.94), the quantities −U Q
and |V |
Q 2 + U 2 may be interpreted as specifying the preferential orientation and
ellipticity of the vibration ellipse. Unlike the Stokes parameters, these quantities are
not additive. In view of the rotation transformation rule (1.97), P, PL , and PC are
invariant with respect to rotations of the reference frame around the direction of
propagation. When U = 0, the ratio
PQ = −
Q
I
(1.114)
is also called the degree of linear polarization (or the signed degree of linear polarization). PQ is positive when the vibrations of the electric vector in the ϕ - direction
(i.e., the direction perpendicular to the meridional plane of the beam) dominate those
in the ϑ - direction and is negative otherwise. The standard polarimetric analysis of a
general quasi-monochromatic beam with Stokes parameters I, Q, U, and V is summarized in Fig. 1.6 (after Hovenier et al. 2005).
30
Scattering, Absorption, and Emission of Light by Small Particles
Further reading
Excellent treatments of classical electrodynamics and optics are provided by Stratton
(1941), Kong (1990), Jackson (1998), and Born and Wolf (1999). The optical properties of bulk matter and their measurement are discussed in Chapters 9 and 10 of
Bohren and Huffman (1983) as well as in the comprehensive handbook edited by
Palik and Ghosh (1997). Several books are entirely devoted to polarization, for example Shurcliff (1962), Clarke and Grainger (1971), Azzam and Bashara (1977), Kliger et al. (1990), Collett (1992), and Brosseau (1998). In Pye (2001), numerous
manifestations of polarization in science and nature are discussed.
Chapter 2
Scattering, absorption, and emission
of electromagnetic radiation by
an arbitrary finite particle
The presence of an object with a refractive index different from that of the surrounding medium changes the electromagnetic field that would otherwise exist in an unbounded homogeneous space. The difference of the total field in the presence of the
object and the original field that would exist in the absence of the object can be
thought of as the field scattered by the object. In other words, the total field is equal
to the vector sum of the incident (original) field and the scattered field.
The angular distribution and polarization of the scattered field depend on the polarization and directional characteristics of the incident field as well as on such properties of the scatterer as its size relative to the wavelength and its shape, composition,
and orientation. Therefore, in practice one usually must solve the scattering problem
anew every time some or all of these input parameters change. It is appropriate, however, to consider first the general mathematical description of the scattering process
without making any detailed assumptions about the scattering object except that it is
composed of a linear and isotropic material. Hence the goal of this chapter is to establish a basic theoretical framework underlying more specific problems discussed in
the following chapters.
2.1
Volume integral equation
Consider a finite scattering object in the form of a single body or a fixed aggregate
embedded in an infinite, homogeneous, linear, isotropic, and nonabsorbing medium
(Fig. 2.1(a)). Mathematically, this is equivalent to dividing all space into two mutually disjoint regions, the finite interior region VINT occupied by the scattering object
and the infinite exterior region VEXT. The region VINT is filled with an isotropic, linear, and possibly inhomogeneous material.
31
32
Scattering, absorption, and emission by small particles
Scattered spherical wave
rˆ = ϑˆ × ϕˆ
Observation point
ϕ̂ = ϕ̂ sca
ϑ̂ = ϑ̂ sca
r
O
ϕ̂inc
z
nˆ sca = rˆ
ϑ̂inc
O
n̂inc = ϑ̂inc × ϕ̂inc
y
x
Incident wave
(a)
(b)
Figure 2.1. Schematic representation of the electromagnetic scattering problem. The unshaded exterior region VEXT is unbounded in all directions and the shaded areas collectively
constitute the interior region VINT .
It is well known that optical properties of bulk substances in solid or liquid phase
are qualitatively different from those of their constituent atoms and molecules when
the latter are isolated. This may cause a problem when one applies the concept of
bulk optical constants to a very small particle because either the optical constants determined for bulk matter provide an inaccurate estimate or the particle is so small that
the entire concept of optical constants loses its validity. We will therefore assume
that the individual bodies forming the scattering object are sufficiently large that they
can still be characterized by optical constants appropriate to bulk matter. According
to Huffman (1988), this implies that each body is larger than approximately 50 Å.
The monochromatic Maxwell curl equations (1.18) and (1.20) describing the
scattering problem can be rewritten as follows:
∇ × E(r ) = iωµ1H (r ) ü
ý
∇ × H (r ) = −iωε 1E(r )þ
∇ × E(r ) = iωµ 2 (r )H (r ) ü
ý
∇ × H (r ) = −iωε 2 (r )E(r )þ
r ∈ VEXT ,
r ∈ VINT ,
(2.1)
(2.2)
where the subscripts 1 and 2 refer to the exterior and interior regions, respectively.
Since the first relations in Eqs. (2.1) and (2.2) yield the magnetic field provided that
2 Scattering, absorption, and emission by an arbitrary finite particle
33
the electric field is known everywhere, we will look for the solution of Eqs. (2.1) and
(2.2) in terms of only the electric field. Assuming that the host medium and the scattering object are nonmagnetic, i.e., µ 2 (r ) ≡ µ1 = µ 0 , where µ 0 is the permeability of
a vacuum, we easily derive the following vector wave equations:
∇ × ∇ × E(r ) − k12 E(r ) = 0,
∇ × ∇ × E(r ) − k 22 (r )E(r )
= 0,
r ∈ VEXT ,
r ∈ VINT ,
(2.3)
(2.4)
where k1 = ω ε 1 µ 0 and k 2 (r ) = ω ε 2 (r ) µ 0 are the wave numbers of the exterior
and interior regions, respectively. The permittivity for the interior region is regarded
as a function of r, to provide for the general case where the scattering object is inhomogeneous. Equations (2.3) and (2.4) can be rewritten as the single inhomogeneous
differential equation
∇ × ∇ × E(r ) − k12 E(r ) = j(r ),
r ∈ VEXT ∪ VINT ,
(2.5)
where
~ 2 (r ) − 1] E(r ),
j(r ) = k12 [m
(2.6a)
~ (r ) = ì1, r ∈ VEXT ,
m
í
î m(r ) = k 2 (r ) k1 = m2 (r ) m1, r ∈ VINT ,
(2.6b)
and m(r ) is the refractive index of the interior relative to that of the exterior. The
forcing function j(r ) obviously vanishes everywhere outside the interior region.
Any solution of an inhomogeneous linear differential equation can be divided into
two parts: (i) a solution of the respective homogeneous equation with the right-hand
side identically equal to zero and (ii) a particular solution of the inhomogeneous
equation. Thus, the first part satisfies the equation
∇ × ∇ × E inc (r ) − k12 E inc (r ) = 0,
r ∈ VEXT ∪ VINT ,
(2.7)
and describes the field that would exist in the absence of the scattering object, i.e., the
incident field. The physically appropriate particular solution of Eq. (2.5) must give
the scattered field E sca (r ) generated by the forcing function j(r ). Obviously, of all
possible particular solutions of Eq. (2.5) we must choose the one that vanishes at large
distances from the scattering object and ensures energy conservation.
To find E sca (r ), we first introduce the free space dyadic Green’s function
t
G (r, r ′ ) as a dyadic (Cartesian tensor) satisfying the differential equation
t
t
t
∇ × ∇ × G (r, r ′ ) − k12 G (r, r ′ ) = I δ(r − r ′ ),
(2.8)
t
where I is the identity dyadic and δ(r − r ′ ) = δ( x − x′ )δ( y − y ′ )δ( z − z ′ ) is the threedimensional Dirac delta function. Note that the result of a dyadic operating on a vector is another vector (see, e.g., Appendix 3 of Van Bladel 1964). This operation may
be thought of as a 3× 3 matrix representing the dyadic multiplying a column matrix
consisting of the initial vector components, thereby producing another column matrix
consisting of the resulting vector components. The components of both vectors must
34
Scattering, absorption, and emission by small particles
be specified in the same coordinate system. From a coordinate-free standpoint, a dyadic can be introduced as a sum of so-called dyads, each dyad being the result of a
dyadic product of two vectors a ⊗ b such that the operation (a ⊗ b) ⋅ c yields the
vector a(b ⋅ c) and the operation c ⋅ (a ⊗ b) yields the vector (c ⋅ a)b. Any dyadic can
be represented as a sum of at most nine dyads. The vector product (a ⊗ b) × c is defined as a dyad a ⊗ (b × c), and c × (a ⊗ b) yields (c × a) ⊗ b. The dot product of dyads a ⊗ b and c ⊗ d yields the dyad (b ⋅ c)(a ⊗ d).
Taking into account that
t
t
∇ × [G (r, r ′ ) ⋅ j(r ′ )] = [∇ × G (r, r ′ )] ⋅ j(r ′ ),
we get
t
t
t
∇ × ∇ × [G (r, r ′ ) ⋅ j(r ′ )] − k12 [G (r, r ′ ) ⋅ j(r ′ )] = I ⋅ j(r ′ )δ(r − r ′ ).
(2.9)
We integrate both sides of this equation over the entire space to obtain
t
dr ′ G (r, r ′ ) ⋅ j(r ′ ) = j(r ).
t
t
(∇ × ∇ × I − k12 I ) ⋅
(2.10)
VINT ∪VEXT
Comparison with Eq. (2.5) now shows that
t
dr ′ G (r, r ′ ) ⋅ j(r ′ ),
E sca (r ) =
r ∈ VINT ∪ VEXT ,
(2.11)
VINT
where we have taken into account that j(r ) vanishes everywhere outside VINT. We
will see in the following section that this particular solution of Eq. (2.5) indeed vanishes at infinity and ensures energy conservation and is therefore the physically appropriate particular solution. Hence, the complete solution of Eq. (2.5) is
t
(2.12)
dr ′ G (r, r ′ ) ⋅ j(r ′ ),
E(r ) = E inc (r ) +
r ∈ VINT ∪ VEXT.
VINT
t
To find the free space dyadic Green’s function G (r, r ′ ), we first express it in
terms of a scalar Green’s function g(r, r ′ ) as follows:
t
æt 1
ö
(2.13)
G (r, r ′ ) = çç I + 2 ∇ ⊗ ∇ ÷÷ g(r, r ′ ).
k1
è
ø
Inserting Eq. (2.13) into Eq. (2.8) and noticing that
∇ × [∇ × (∇ ⊗ ∇)] = ∇ × [(∇ × ∇) ⊗ ∇] = 0,
t
t
∇ × ∇ × ( I g ) = ∇ ⊗ ∇g − I ∇ 2 g ,
we obtain the following differential equation for g:
(∇ 2 + k12 )g(r, r ′ ) = − δ(r − r ′ ).
(2.14)
The well-known solution of this equation representing so-called outgoing waves (i.e.,
satisfying the condition lim g(r, r ′ ) = 0) is
k1 |r − r ′|→ ∞
2 Scattering, absorption, and emission by an arbitrary finite particle
g(r, r ′ ) =
e ik |r −r ′|
4π |r − r ′|
1
35
(2.15)
(e.g., Jackson 1998, p. 427). Hence, Eqs. (2.6), (2.12), (2.13), and (2.15) finally yield
(Shifrin 1968; Saxon 1955b)
t
E(r ) = E inc (r ) + k12
dr ′ G (r, r ′ ) ⋅ E(r ′ ) [m 2 (r ′ ) − 1]
VINT
æt 1
ö
= E inc (r ) + k12 çç I + 2 ∇ ⊗ ∇ ÷÷ ⋅
k1
è
ø
dr ′ [m 2 (r ′ ) − 1] E(r ′ )
VINT
e ik | r − r ′ |
,
4π |r − r ′|
1
r ∈ VINT ∪ VEXT.
(2.16)
Equation (2.16) expresses the total electric field everywhere in space in terms of
the incident field and the total field inside the scattering object. Since the latter is not
known in general, one must solve Eq. (2.16) either numerically or analytically. As a
first step, the internal field can be approximated by the incident field. This is the gist
of the so-called Rayleigh–Gans approximation otherwise known as the Rayleigh–Debye or Born approximation (van de Hulst 1957; Ishimaru 1997). The total field computed in the Rayleigh-Gans approximation can be substituted in the integral on the
right-hand side of Eq. (2.16) in order to compute an improved approximation, and this
iterative process can be continued until the total field converges within a given numerical accuracy. Although this procedure can be rather involved, it shows that in the
final analysis the total electric field can be expressed in terms of the incident field as
follows:
t
t
E(r ) = E inc (r ) +
dr ′ G (r, r ′ ) ⋅
dr ′′ T (r ′, r ′′ ) ⋅ E inc (r ′′ ), r ∈ VINT ∪ VEXT ,
VINT
VINT
(2.17)
t
where T is the so-called dyadic transition operator (Tsang et al. 1985). Substituting
t
Eq. (2.17) in Eq. (2.16), we derive the following integral equation for T :
t
t
T (r, r ′ ) = k12 [m 2 (r ) − 1]δ(r − r ′ ) I
t
t
dr ′′ G (r, r ′′ ) ⋅ T (r ′′, r ′ ),
+ k12 [m 2 (r ) − 1]
r, r ′ ∈ VINT.
(2.18)
VINT
Equations of this type appear in the quantum theory of scattering and are called
Lippmann-Schwinger equations (Lippmann and Schwinger 1950; Newton 1966).
2.2
Scattering in the far-field zone
Let us now choose an arbitrary point O close to the geometrical center of the scattering object as the common origin of all position (radius) vectors (Figs. 2.1(a), (b)).
Usually one is interested in calculating the scattered field in the so-called far-field
36
Scattering, absorption, and emission by small particles
zone. Specifically, let us assume that k1r o 1 and that r is much greater than any linear dimension of the scattering object (r o r ′ for any r ′ ∈ VINT ). Since
|r − r ′| = r 1 − 2
r ′2
rˆ ⋅ r ′ r ′ 2
+ 2 ≈ r − rˆ ⋅ r ′ +
,
r
2r
r
(2.19)
where rˆ = r r is the unit vector in the direction of r (Fig. 2.1(b)), we have
g(r, r ′ ) ≈
e ik r −ik rˆ ⋅r ′
e
,
4π r
1
1
where it is also assumed that k1r ′ 2 2r n 1. Therefore,
t
t
e ik r −ik rˆ ⋅r ′
G (r, r ′ ) ≈ ( I − rˆ ⊗ rˆ )
e
.
4π r
1
(2.20)
1
In deriving Eq. (2.20), we have taken into account that in spherical coordinates, defined in Section 1.3, centered at the origin,
∇ = rˆ
∂ ˆ1 ∂
∂
1
+ϑ
+ ϕˆ
,
∂r
r ∂ϑ
r sin ϑ ∂ϕ
(2.21)
where the order of operator components relative to the unit basis vectors is essential
because r̂, ϑ̂, and ϕ̂ depend on ϑ and ϕ . Hence,
E sca (r ) ≈
e ik r k12 t
( I − rˆ ⊗ rˆ ) ⋅
r 4π
1
dr ′ [m 2 (r ′ ) − 1] E(r ′ )e −ik rˆ ⋅r ′.
1
(2.22)
VINT
This important formula shows that the scattered field at a large distance from the
object behaves as an outgoing transverse spherical wave. Specifically, since the
identity dyadic in the spherical coordinate system centered at the origin is given by
t
I = rˆ ⊗ rˆ + ϑˆ ⊗ ϑˆ + ϕˆ ⊗ ϕˆ ,
t
the factor I − rˆ ⊗ rˆ = ϑˆ ⊗ ϑˆ + ϕˆ ⊗ ϕˆ ensures that the scattered wave in the far-field
zone is transverse, i.e., the electric field vector is always perpendicular to the direction of propagation r̂ :
rˆ ⋅ E sca (r ) = 0.
(2.23)
Hence, only the ϑ - and ϕ - components of the electric vector of the scattered field
are non-zero. Furthermore, the scattered field decays inversely with distance r from
the scattering object. Equation (2.22) can be rewritten in the form
E sca (r ) =
e ik r sca
E1 (rˆ ),
r
1
rˆ ⋅ E1sca (rˆ ) = 0,
(2.24)
where the vector E1sca (rˆ ) is independent of r and describes the angular distribution of
the scattered radiation in the far-field zone. Obviously, this solution also obeys the
2 Scattering, absorption, and emission by an arbitrary finite particle
37
energy conservation law by making the total energy flux across a spherical surface of
radius r independent of r.
Assuming that the incident field is a plane electromagnetic wave given by
ˆ inc ⋅ r )
E inc (r ) = E inc
0 exp(ik1n
(2.25)
and using Eq. (2.17), we derive for the far-field zone
e ik r t sca inc
E sca (r nˆ sca ) =
A(nˆ , nˆ ) ⋅ E inc
0 ,
r
1
t
where nˆ sca = rˆ (Fig. 2.1(b)) and the scattering dyadic A is given by
t
1 t
A(nˆ sca , nˆ inc ) =
( I − nˆ sca ⊗ nˆ sca ) ⋅
dr ′ exp(−ik1nˆ sca ⋅ r ′ )
4π
V
t
×
dr ′′ T (r ′, r ′′ )exp(ik1nˆ inc ⋅ r ′′ ).
(2.26)
INT
(2.27)
VINT
The elements of the scattering dyadic have the dimension of length.
Equation (2.17) shows that if E1inc (r ) and E inc
2 (r ) are two different incident fields
sca
sca
and E1 (r ) and E 2 (r ) are the corresponding scattered fields, then E1sca (r ) + E sca
2 (r )
inc
inc
is the scattered field corresponding to the incident field E1 (r ) + E 2 (r ). This result
is, of course, a consequence of the linearity of Maxwell’s equations (2.1) and (2.2)
and constitutive relations (1.8)–(1.10) and a manifestation of the well-known principle of superposition: if two electromagnetic fields satisfy the Maxwell equations, then
t
their sum also satisfies these equations. Therefore, although the scattering dyadic A
describes the scattering of a plane electromagnetic wave, it can be used to compute
the scattering of any incident field as long as the latter can be expanded in elementary
plane waves.
It follows from Eqs. (2.23) and (2.27) that
t
nˆ sca ⋅ A(nˆ sca , nˆ inc ) = 0.
(2.28)
However, because the incident field given by Eq. (2.25) is a transverse wave with
electric vector perpendicular to the direction of propagation, the dot product
t
A(nˆ sca , nˆ inc ) ⋅ nˆ inc is not defined by Eq. (2.26). To complete the definition, we take
this product to be zero:
t
A(nˆ sca , nˆ inc ) ⋅nˆ inc = 0,
(2.29)
which means that one must retain only the part of the expression on the right-hand
side of Eq. (2.27) that is transverse to the incidence direction. As a consequence of
Eqs. (2.28) and (2.29), only four out of the nine components of the scattering dyadic
are independent. It is therefore convenient to formulate the scattering problem in the
spherical coordinate system centered at the origin and to introduce the 2 × 2 so-called
amplitude scattering matrix S, which describes the transformation of the ϑ - and
ϕ - components of the incident plane wave into the ϑ - and ϕ - components of the
scattered spherical wave:
38
Scattering, absorption, and emission by small particles
é Eϑsca (r nˆ sca )ù e ik r
é E0inc
ϑ ù
sca
inc
ˆ
ˆ
S
(
n
,
n
)
=
ê sca
ú
ê inc ú.
sca
r
êë Eϕ (r nˆ )úû
êë E0ϕ úû
1
(2.30)
The amplitude scattering matrix depends on the directions of incidence and scattering
as well as on the size, morphology, composition, and orientation of the scattering object with respect to the coordinate system. As will be discussed in Section 2.11, it
also depends on the choice of origin of the coordinate system inside the scattering
object. If known, the amplitude scattering matrix gives the scattered and thus the total
field, thereby providing a complete description of the scattering pattern in the far-field
zone. The elements of the amplitude scattering matrix have the dimension of length
and are expressed in terms of the scattering dyadic as follows:
t
S11 = ϑˆ sca ⋅ A ⋅ ϑˆ inc ,
(2.31)
t
S12 = ϑˆ sca ⋅ A ⋅ ϕˆ inc ,
(2.32)
t
S 21 = ϕˆ sca ⋅ A ⋅ ϑˆ inc ,
(2.33)
t
S 22 = ϕˆ sca ⋅ A ⋅ ϕˆ inc.
(2.34)
We have pointed out in Section 1.3 that when a wave propagates along the z-axis,
the ϑ - and ϕ - components of the electric field vector are determined by the specific
choice of meridional plane. Therefore, the amplitude scattering matrix explicitly depends on ϕ inc and ϕ sca even when ϑ inc = 0 or π and/or ϑ sca = 0 or π .
2.3
Reciprocity
A fundamental property of the scattering dyadic is the reciprocity relation, which is a
manifestation of the symmetry of the scattering process with respect to an inversion
of time (Saxon 1955a). To derive the reciprocity relation, we first consider the scattering of a spherical incoming wave by an arbitrary finite object embedded in an infinite, homogeneous, nonabsorbing medium. In the far-field zone of the object, the
total electric field is the sum of the incoming and scattered spherical waves:
E(r rˆ ) =
e −ik r inc
e ik r sca
E (rˆ ) +
E (rˆ ),
r
r
1
1
(2.35)
where E inc (rˆ ) and E sca (rˆ ) are independent of r and
rˆ ⋅ E inc (rˆ ) = 0,
(2.36)
rˆ ⋅ E sca (rˆ ) = 0.
(2.37)
Equation (2.37) follows from Eq. (2.24), whereas Eq. (2.36) follows from the divergence condition
∇ ⋅ E(r ) = 0
(2.38)
2 Scattering, absorption, and emission by an arbitrary finite particle
39
(Eq. (1.17) with ε1 = constant) and the following relations:
∇ ⋅ ( f a) = f ∇ ⋅ a + (∇f ) ⋅ a,
(2.39)
e ± ik1r
æ1
ö e ± ik1r ˆ
= − ç m ik1 ÷
r,
r
èr
ø r
∇
(2.40)
é e −ik r inc ù e −ik r
ö e −ik r ˆ inc ˆ
æ1
∇⋅ê
E (rˆ )ú =
∇ ⋅ E inc (rˆ ) − ç + ik1 ÷
r ⋅ E (r ) = 0,
r
ø r
èr
ë r
û
(2.41)
∇ ⋅ E inc (rˆ ) = O(r −1).
(2.42)
1
1
1
r →∞
The latter is a consequence of Eq. (2.21) and the fact that E inc (rˆ ) is independent of r.
Because of the linearity of the Maxwell equations and by analogy with Eq. (2.26),
the outgoing spherical wave must be linearly related to the incoming spherical wave.
Following Saxon (1955a), we express this relationship in terms of the so-called scatt
tering tensor S as follows:
t
drˆ ′ S (rˆ , rˆ ′ ) ⋅ E inc (−rˆ ′ ),
E sca (rˆ ) = −
(2.43)
4π
where
drˆ ′ =
4π
2π
0
dϕ ′
π
dϑ ′ sinϑ ′.
(2.44)
0
In view of Eq. (2.37), we have
t
rˆ ⋅ S (rˆ , rˆ ′ ) = 0.
(2.45)
t
Since E inc (rˆ ) is transverse, the product S (rˆ , rˆ ′ ) ⋅ rˆ ′ remains undefined by Eq. (2.43).
Therefore, we will complete the definition of the scattering tensor by taking this
product to be zero:
t
S (rˆ , rˆ ′ ) ⋅ rˆ ′ = 0.
(2.46)
t
As a consequence of Eqs. (2.45) and (2.46), S has only four independent components.
The derivation of the reciprocity relation for the scattering tensor starts from the
fact that if E1 and E 2 are any two solutions of the source-free Maxwell equations
(but with the same harmonic time dependence) then
r2
drˆ rˆ ⋅ {E 2 (r rˆ ) × [∇ × E1(r rˆ )] − E1(r rˆ ) × [∇ × E 2 (r rˆ )]} = 0.
4π
r →∞
(2.47)
Indeed, using Eqs. (1.26), (2.1), and (2.2), it can easily be established that
∇ ⋅ (E 2 × H1 − E1 × H 2 ) vanishes identically everywhere in space. Integrating ∇ ⋅ (E 2
× H1 − E1 × H 2 ) over all space and applying the Gauss theorem then yields Eq. (2.47).
We now take E1 and E 2 to be superpositions of incoming and outgoing spherical
waves:
Scattering, absorption, and emission by small particles
40
e −ik r inc
e ik r sca
E j (rˆ ) +
E j (rˆ ),
r →∞
r
r
E j (r rˆ ) =
1
1
j = 1, 2.
(2.48)
Taking into account Eq. (1.40), (2.36), (2.37), and (2.40) and the formulas
∇ × ( f a) = (∇f ) × a + f (∇ × a),
(2.49)
inc,sca
∇ × E1,2
(rˆ ) = O(r −1),
(2.50)
r →∞
cf. Eq. (2.21), we derive the following after some algebra:
ˆ ) ⋅ E1sca (rˆ ) − E1inc (rˆ ) ⋅ E sca
ˆ )] = 0.
drˆ [E inc
2 (r
2 (r
(2.51)
4π
Using Eq. (2.43) to express the outgoing waves in terms of the incoming waves, we
then have
t
t
ˆ ) ⋅ S (rˆ , rˆ ′ ) ⋅ E1inc (−rˆ ′ ) − E1inc (rˆ ) ⋅ S (rˆ , rˆ ′ ) ⋅ E inc
ˆ ′ )] = 0.
drˆ ′ [E inc
2 (r
2 ( −r
drˆ
4π
(2.52)
4π
Replacing r̂ by −rˆ ′ and rˆ ′ by −r̂ in the last term and transposing the tensor product
according to the identity
t
t
a ⋅ B ⋅ c = c ⋅ BT ⋅ a
we derive
t
t
ˆ ) ⋅ [ S (rˆ , rˆ ′ ) − S T (−rˆ ′, − rˆ )] ⋅ E1inc (−rˆ ′ ) = 0,
drˆ ′ E inc
2 (r
drˆ
4π
(2.53)
4π
where T denotes the transposed tensor:
t
t
( S T ) ij = S ji.
(2.54)
Since E1inc and Einc
2 are arbitrary, we finally have
t
t
S (rˆ , rˆ ′ ) = S T (−rˆ ′, − rˆ ).
(2.55)
This is the reciprocity condition for the scattering tensor.
It should be remarked that in deriving Eq. (2.47) we assumed, as almost everywhere else in this book, that the permeability, permittivity, and conductivity are scalars. However, it is easily checked that Eq. (2.47) and thus the reciprocity condition
(2.55) remain valid even when the permeability, permittivity, and conductivity of the
scattering object are tensors, provided that all these tensors are symmetric. If any of
these tensors is not symmetric, then Eq. (2.55) may become invalid (Dolginov et al.
1995; Lacoste and van Tiggelen 1999).
We now use Eq. (2.55) to derive the reciprocity relation for the scattering dyadic
t
A by considering the case in which the scattering object is illuminated by a plane
wave incident along the direction nˆ inc. As follows from Eqs. (2.24) and (2.25), the
total electric field in the far-field zone is given by
2 Scattering, absorption, and emission by an arbitrary finite particle
ˆ inc ⋅ nˆ sca ) + E1sca (nˆ sca )
E(r nˆ sca ) = E inc
0 exp(ik1r n
e ik r
.
r
1
41
(2.56)
Representing the incident plane wave as a superposition of incoming and outgoing
spherical waves,
i 2π é inc
e − ik r
e ik r ù
δ(nˆ + nˆ sca )
− δ(nˆ inc − nˆ sca )
ê
ú
k r →∞ k
r
r û
1 ë
1
exp(ik1r nˆ inc ⋅ nˆ sca ) =
1
(2.57)
1
(see Appendix A), where
δ(nˆ inc ± nˆ sca ) = δ(cosϑ inc ± cosϑ sca ) δ(ϕ inc ± ϕ sca )
(2.58)
is the solid-angle Dirac delta function, we derive
i 2π inc
e − ik r
E 0 δ(nˆ inc + nˆ sca )
k r →∞ k
r
1
1
E(r nˆ sca ) =
1
é
ù e ik r
i 2π
+ êE1sca (nˆ sca ) −
δ(nˆ inc − nˆ sca )E inc
.
0 ú
k1
ë
û r
1
(2.59)
Considering this a special form of Eq. (2.35) and recalling the definition of the scattering tensor, Eq. (2.43), we have
E1sca (nˆ sca ) =
t
i 2π
[δ(nˆ inc − nˆ sca )E inc
−
S
(nˆ sca , nˆ inc ) ⋅ E inc
0
0 ].
k1
(2.60)
It now follows from the definition of the scattering dyadic, Eqs. (2.26), (2.28), and
(2.29), that
t
t
i 2π t
A(nˆ sca , nˆ inc ) =
[( I − nˆ inc ⊗ nˆ inc ) δ(nˆ inc − nˆ sca ) − S (nˆ sca , nˆ inc )].
k1
(2.61)
Finally, from Eqs. (2.55) and (2.61) we derive the reciprocity relation for the scattering dyadic:
t
t
A(−nˆ inc , − nˆ sca ) = AT (nˆ sca , nˆ inc ).
(2.62)
The reciprocity relation for the amplitude scattering matrix follows from Eqs.
(2.31)–(2.34) and (2.62) and the unit vector identities
ϑˆ (−nˆ ) = ϑˆ (nˆ ),
ϕˆ (−nˆ ) = − ϕˆ (nˆ ).
(2.63)
Simple algebra gives
é S11(nˆ sca , nˆ inc ) − S 21(nˆ sca , nˆ inc )ù
S(−nˆ inc , − nˆ sca ) = ê
ú.
êë− S12 (nˆ sca , nˆ inc ) S 22 (nˆ sca , nˆ inc ) úû
(2.64)
An interesting consequence of reciprocity is the so-called backscattering theorem,
which directly follows from Eq. (2.64) after substituting nˆ inc = nˆ and nˆ sca = −nˆ :
Scattering, absorption, and emission by small particles
42
S 21(−nˆ , nˆ ) = − S12 (−nˆ , nˆ )
(2.65)
(van de Hulst 1957, Section 5.32).
Because of the universal nature of reciprocity, Eqs. (2.62), (2.64), and (2.65) are
important tests in computations or measurements of light scattering by small particles;
violation of reciprocity means that the computations or measurements are incorrect or
inaccurate. Alternatively, the use of reciprocity can substantially shorten the required
computer time or reduce the measurement effort because one may calculate or measure light scattering for only half of the scattering geometries and then use Eqs. (2.62)
and (2.64) for the reciprocal geometries. Reciprocity plays a fundamental role in the
phenomenon of coherent backscattering of light from discrete random media discussed in Section 3.4 (e.g., Mishchenko 1992b; van Tiggelen and Maynard 1997).
2.4
Reference frames and particle orientation
It is often convenient to specify the orientation of the scattering object using the same
fixed reference frame that is used to specify the directions and states of polarization of
the incident and scattered waves. In what follows, we will refer to this reference
frame as the laboratory coordinate system and denote it by L. Although the spatial
orientation of the laboratory coordinate system is, in principle, arbitrary, it can often
be chosen in such a way that it most adequately represents the geometry of the scattering medium or the physical mechanism of particle orientation. In order to describe
the orientation of the scattering object with respect to the laboratory reference frame,
we introduce a right-handed coordinate system P affixed to the particle and having the
same origin inside the particle as L. This coordinate system will be called the particle
reference frame. The orientation of the particle with respect to L is specified by three
Euler angles of rotation, α , β , and γ , which transform the laboratory coordinate system
L{x, y, z} into the particle coordinate system P{x′, y ′, z ′ }, as shown in Fig. 2.2. The
three consecutive Euler rotations are performed as follows:
●
●
●
rotation of the laboratory coordinate system about the z-axis through an angle
α ∈ [0, 2π ), reorienting the y-axis in such a way that it coincides with the line of
nodes (i.e., the line formed by the intersection of the xy- and x′y ′ - planes);
rotation about the new y-axis through an angle β ∈ [0, π ];
rotation about the z ′- axis through an angle γ ∈ [0, 2π ).
An angle of rotation is positive if the rotation is performed in the clockwise direction
when one is looking in the positive direction of the rotation axis.
As we will see in Chapters 5 and 6, most of the available analytical and numerical
techniques assume that (or become especially efficient when) the scattering problem
is solved in the particle reference frame with coordinate axes directed along the axes
of particle symmetry. This implies that the incidence and scattering directions and
polarization reference planes must also be specified with respect to the particle refer-
2 Scattering, absorption, and emission by an arbitrary finite particle
43
z
z′
x
β
y′
α
γ
β
y
Line of nodes
x′
Figure 2.2. Euler angles of rotation α, β , and γ transforming the laboratory coordinate
system L{x, y, z} into the particle coordinate system P{x′, y′, z′}.
ence frame. Therefore, in order to solve the scattering problem with respect to the
laboratory reference frame, one must first determine the illumination and scattering
directions with respect to the particle reference frame for a given orientation of the
particle relative to the laboratory reference frame, then solve the scattering problem in
the particle reference frame, and finally perform the backward transition to the laboratory reference frame. In this section we derive general formulas describing this
procedure (Mishchenko 2000).
Consider a monochromatic plane electromagnetic wave with electric field vector
inc inc
ˆ inc
ˆ L ) exp(ik1nˆ inc ⋅ r )
E inc (r ) = ( E0inc
ϑL ϑ L + E 0ϕL ϕ
(2.66)
incident upon a nonspherical particle in a direction nˆ inc , where r is the position (radius) vector connecting the origin of the laboratory coordinate system and the observation point and the index L labels unit vectors and electric field vector components
computed in the laboratory reference frame. In the far-field region, the scattered field
vector components are given by
é Eϑsca
é E0inc
ˆ sca )ù
exp(ik1r ) L sca inc
L (r n
ϑL ù
ˆ
ˆ
=
S
(
n
,
n
;
α
,
β
,
γ
)
ê sca
ú
ê inc ú,
sca
r
ëê E0ϕL ûú
ëê Eϕ L (r nˆ )ûú
(2.67)
where S L is the 2 × 2 amplitude scattering matrix in the laboratory reference frame.
The amplitude scattering matrix depends on the directions of incidence and scattering
as well as on the orientation of the scattering particle with respect to the laboratory
reference frame as specified by the Euler angles of rotation α , β , and γ .
Assume now that one of the available analytical or numerical techniques can be
efficiently used to find the amplitude scattering matrix with respect to the particle
44
Scattering, absorption, and emission by small particles
reference frame. This matrix will be denoted by S P and relates the incident and scattered field vector components computed in the particle reference frame for the same incidence and scattering directions:
inc
ˆ sca )ù
é Eϑsca
exp(ik1r ) P sca inc é E0ϑP ù
P (r n
ˆ , nˆ ) ê
=
S
(
n
ê sca
ú
ú.
sca
inc
r
ëê Eϕ P (r nˆ )ûú
ëê E0ϕ P ûú
(2.68)
The amplitude scattering matrix with respect to the laboratory reference frame can be
expressed in terms of the matrix S P as follows. Denote by t a 2 × 2 matrix that
transforms the electric field vector components of a transverse electromagnetic wave
computed in the laboratory reference frame into those computed in the particle reference frame:
é EϑP (ϑ P , ϕ P ) ù
é EϑL (ϑ L , ϕ L )ù
ê
ú = t (nˆ ; α , β , γ ) ê
ú,
ëê Eϕ P (ϑ P , ϕ P )ûú
ëê Eϕ L (ϑ L , ϕ L )ûú
(2.69)
where n̂ is a unit vector in the direction of light propagation; (ϑ L , ϕ L ) and (ϑ P , ϕ P )
specify this direction with respect to the laboratory and particle reference frames, respectively. The t matrix depends on n̂ as well as on the orientation of the particle
relative to the laboratory reference frame, specified by the Euler angles α , β , and γ .
The inverse transformation is
é EϑL (ϑ L , ϕ L )ù
é EϑP (ϑ P , ϕ P ) ù
−1
ê
ú = t (nˆ ; α , β , γ ) ê
ú,
êë Eϕ L (ϑ L , ϕ L )úû
êë Eϕ P (ϑ P , ϕ P )úû
(2.70)
é1 0ù
t (nˆ ; α , β , γ ) t −1(nˆ ; α , β , γ ) = ê
ú.
ë0 1 û
(2.71)
where
We then easily derive
S L (ϑLsca , ϕ Lsca; ϑLinc, ϕ Linc; α , β , γ ) = t −1(nˆ sca; α , β , γ )
× S P (ϑ Psca , ϕ Psca ; ϑ Pinc , ϕ Pinc ) t (nˆ inc; α , β , γ ).
(2.72)
To determine the matrix t, we proceed as follows. Denote by α a 3× 2 matrix
that transforms the ϑ and ϕ components of the electric field vector into its x, y, and z
components,
é Ex ù
é Eϑ ù
ê ú
=
α
(
ϑ
,
ϕ
)
E
ê ú,
y
ê ú
êë Eϕ úû
ê Ez ú
ë û
(2.73)
and by β a 3× 3 matrix that expresses the x, y, and z components of a vector in the
particle coordinate system in terms of the x, y, and z components of the same vector in
the laboratory coordinate system,
2 Scattering, absorption, and emission by an arbitrary finite particle
é E xP ù
é E xL ù
ê
ú
ê
ú
ê E yP ú = β(α , β , γ ) ê E yL ú.
ê E zP ú
ê E zL ú
ë
û
ë
û
45
(2.74)
We then have
t (nˆ ; α , β , γ ) = α −1(ϑ P , ϕ P ) β(α , β , γ ) α(ϑ L , ϕ L ),
(2.75)
where α −1(ϑ P , ϕ P ) is a suitable left inverse of α(ϑ P , ϕ P ).
The matrices entering the right-hand side of Eq. (2.75) are as follows (Arfken and
Weber 1995, pp. 118 and 189):
écosϑ cosϕ
ê
α(ϑ , ϕ ) = ê cosϑ sinϕ
ê − sinϑ
ë
écosϑ cosϕ
α −1(ϑ , ϕ ) = ê
ë − sinϕ
− sinϕ ù
ú
cosϕ ú,
0 úû
(2.76)
cosϑ sinϕ
cosϕ
é cos α cosβ cosγ − sin α sinγ
ê
β(α , β , γ ) = ê− cos α cosβ sinγ − sin α cosγ
ê
cos α sinβ
ë
− sinϑ ù
ú,
0 û
sin α cosβ cosγ + cos α sinγ
− sin α cosβ sinγ + cos α cosγ
sin α sinβ
(2.77)
− sinβ cosγ ù
ú
sinβ sinγ ú.
cosβ úû
(2.78)
To express the angles ϑ P and ϕ P in terms of the angles ϑ L and ϕ L , we rewrite Eq.
(2.74) as
ésinϑ P cosϕ P ù
ésinϑ L cosϕ L ù
ê
ú
ê
ú
ê sinϑ P sinϕ P ú = β(α , β , γ ) ê sinϑ L sinϕ L ú,
ê cosϑ P ú
ê cosϑ L ú
ë
û
ë
û
(2.79)
where (ϑ L , ϕ L ) and (ϑ P , ϕ P ) are spherical angular coordinates of an arbitrary unit
vector in the laboratory and particle reference frames, respectively. Equations (2.78)
and (2.79) and simple algebra then give
cosϑ P = cosϑ L cosβ + sinϑ L sinβ cos(ϕ L − α ),
cosϕ P =
1
[cosβ cosγ sinϑ L cos(ϕ L − α )
sinϑ P
+ sinγ sinϑ L sin(ϕ L − α ) − sinβ cosγ cosϑ L ],
sinϕ P =
(2.80)
(2.81)
1
[− cosβ sinγ sinϑ L cos(ϕ L − α )
sinϑ P
+ cosγ sinϑ L sin(ϕ L − α ) + sinβ sinγ cosϑ L ].
(2.82)
One easily verifies that if α = 0, β = 0, and γ = 0 (i.e., the particle reference
Scattering, absorption, and emission by small particles
46
frame coincides with the laboratory reference frame), then ϑ P = ϑ L , ϕ P = ϕ L ,
é1 0ù
t (nˆ ; α = 0, β = 0, γ = 0) ≡ ê
ú,
ë0 1 û
(2.83)
S L (ϑ Lsca , ϕ Lsca ; ϑ Linc , ϕ Linc; 0, 0, 0) = S P (ϑ Psca , ϕ Psca ; ϑ Pinc , ϕ Pinc ).
(2.84)
and
For rotationally symmetric particles, it is often advantageous to choose the particle
coordinate system such that its z-axis is directed along the axis of particle symmetry.
In this case the orientation of the particle with respect to the laboratory coordinate
system is independent of the Euler angle γ , so that we can set γ = 0 and get instead
of Eqs. (2.78), (2.81), and (2.82)
écos α cosβ
ê
β(α , β , γ = 0) = ê − sin α
ê cos α sinβ
ë
sin α cosβ
cos α
sin α sinβ
− sinβ ù
ú
0 ú,
cosβ úû
(2.85)
cos ϕ P =
1
[cosβ sinϑ L cos(ϕ L − α ) − sinβ cosϑ L ],
sinϑ P
(2.86)
sinϕ P =
sinϑ L sin(ϕ L − α )
.
sinϑ P
(2.87)
In summary, the numerical scheme for computing the amplitude scattering matrix
for given ϑ Linc , ϕ Linc , ϑ Lsca , ϕ Lsca , α , β , and γ is as follows:
●
●
calculation of ϑ Pinc , ϕ Pinc , ϑ Psca , and ϕ Psca via Eqs. (2.80)–(2.82);
calculation of the matrix β(α , β , γ ) via Eq. (2.78);
●
calculation of the matrices α(ϑ Linc , ϕ Linc ), α(ϑ Lsca , ϕ Lsca ), α −1(ϑ Pinc , ϕ Pinc ), and
α −1(ϑ Psca , ϕ Psca ) via Eqs. (2.76) and (2.77);
calculation of the matrices t (nˆ inc; α , β , γ ) and t −1(nˆ sca ; α , β , γ ) via Eq.
(2.75);
●
●
calculation of the matrix S P (ϑ Psca , ϕ Psca ; ϑ Pinc , ϕ Pinc ) using one of the available
analytical or numerical techniques;
●
calculation of the matrix S L (ϑ Lsca , ϕ Lsca ; ϑ Linc , ϕ Linc; α , β , γ ) via Eq. (2.72).
We finally remark that because the particle reference frame can, in principle, be
chosen arbitrarily, Eq. (2.72) can be considered as a general rotation transformation
law expressing the amplitude scattering matrix in the original coordinate system in
terms of the amplitude scattering matrix computed in a rotated coordinate system.
2.5
Poynting vector of the total field
Although the knowledge of the amplitude scattering matrix provides a complete de-
2 Scattering, absorption, and emission by an arbitrary finite particle
47
scription of the monochromatic scattering process in the far-field zone, measurement
of the amplitude scattering matrix is a very complicated experimental problem involving the determination of both the amplitude and the phase of the incident and
scattered waves. Measuring the phase is especially difficult, and only a handful of
such experiments have been performed, all using the microwave analog technique
(Gustafson 2000). The majority of other experiments have dealt with quasimonochromatic rather than monochromatic light and involved measurements of derivative quantities having the dimension of energy flux rather than the electric field
itself. It is therefore useful to characterize the scattering process using quantities that
are easier to measure and are encountered more often, even though they may provide
a less complete description of the scattering pattern in some cases. These quantities
will be introduced in this and the following sections.
We begin by writing the time-averaged Poynting vector áS(r )ñ at any point in the
far-field zone as the sum of three terms:
áS(r )ñ = 12 Re[ E(r ) × H ∗(r )] = áS inc (r )ñ + áS sca (r )ñ + áS ext (r )ñ,
(2.88)
áS inc (r )ñ = 12 Re{E inc (r ) × [H inc (r )]∗ }
(2.89)
áS sca (r )ñ = 12 Re{E sca (r ) × [H sca (r )]∗ }
(2.90)
where
and
are Poynting vectors associated with the incident and the scattered fields, respectively, whereas
áS ext (r )ñ = 12 Re{E inc (r ) × [H sca (r )]∗ + E sca (r ) × [H inc (r )]∗ }
(2.91)
can be interpreted as a term caused by interaction between the incident and the scattered fields. Let us consider a scattering object illuminated by a plane electromagnetic
wave. Recalling Eqs. (1.36), (1.38), (1.42), (2.25), and (2.57), we have for the incident wave in the far-field zone of the scattering particle
ˆ inc ⋅ r )
E inc (r ) = E inc
0 exp(ik1n
i 2π
k1r → ∞ k
1
=
H inc (r ) =
ˆ inc = 0,
E inc
0 ⋅n
(2.92)
ε1
exp(ik1nˆ inc ⋅ r ) nˆ inc × E inc
0
µ0
i 2π
k1r → ∞ k
1
=
é inc
e −ik1r
e ik1r ù inc
inc
ˆ
ˆ
ˆ
ˆ
δ
(
n
+
r
)
−
δ
(
n
−
r
)
ê
ú E0 ,
r
r û
ë
é inc
e −ik1r
e ik1r ù
inc
ˆ
ˆ
ˆ
ˆ
δ
(
n
+
r
)
−
δ
(
n
−
r
)
ê
ú
r
r û
ë
ε1 inc
nˆ × E inc
0 ,
µ0
(2.93)
where r = r rˆ is the radius vector connecting the particle and the observation point.
The first relation of Eq. (2.1) and Eqs. (2.23), (2.24), (2.40), and (2.49)–(2.50) give
for the scattered wave:
Scattering, absorption, and emission by small particles
r2
48
D
et
ec
to
∆S
Incident plane wave
Scattered spherical wave
n̂ sca
O
n̂ inc
Detector 1
Figure 2.3. The response of the collimated detector depends on the line of sight.
e ik1r sca
E1 (rˆ ),
k1r → ∞ r
E sca (r ) =
E1sca (rˆ ) ⋅ rˆ = 0,
ε1 e ik r
rˆ × E1sca (rˆ ).
µ0 r
1
H sca (r ) =
k1r → ∞
(2.94)
(2.95)
Consider now a well-collimated detector of electromagnetic radiation placed at a
distance r from the particle in the far-field zone, with its surface ∆S aligned normal
to and centered on the straight line extending from the particle in the direction of the
unit vector n̂ inc (Fig. 2.3). We assume that the dimension of the detector surface is
much greater than any dimension of the scattering object and the wavelength but
much smaller than r. Furthermore, we assume that ∆S r 2 is smaller than the detector solid-angle field of view Ω so that all radiation scattered by the particle and impinging on ∆S is detected. Obviously, the term áS inc (r )ñ does not contribute to the
detected signal unless rˆ = nˆ inc. From Eqs. (2.88)–(2.95), it is straightforward to show
that the total electromagnetic power received by the detector is
W∆S (nˆ sca ) =
dS rˆ ⋅ áS(r )ñ
∆S
≈
1
2
ε1 ∆S sca sca 2
|E1 (nˆ ) |
µ0 r 2
when nˆ sca ≠ nˆ inc , whereas for the exact forward-scattering direction
(2.96)
2 Scattering, absorption, and emission by an arbitrary finite particle
W∆S (nˆ inc ) =
49
dS nˆ inc ⋅ áS(r )ñ
∆S
=
ε
1
2
∆S 1 |Einc
0 | +
µ0
2
≈
≈
dS nˆ inc ⋅ [áSsca (r )ñ + áS ext (r )ñ ]
∆S
1
1 sca inc 2 ù 2
ε é
2
ˆ )| ú + r
∆S 1 ê|Einc
0 | + 2 |E1 (n
2
r
µ0 ë
û
1 sca inc 2 ù 2π
ε é
1
2
ˆ )| ú −
∆S 1 ê|Einc
0 | + 2 |E1 (n
µ0 ë
r
2
û k1
=
ε
1
2π
2
∆S 1 |E inc
0 | −
µ0
2
k1
drˆ nˆ inc ⋅ áS ext (r r )ñ
∆Ω
ε1
∗
Im[ E1sca (nˆ inc) ⋅ Einc
0 ]
µ0
ε1
∗
−2
Im[E1sca (nˆ inc ) ⋅ E inc
0 ] + O ( r ),
µ0
(2.97)
where ∆Ω = ∆S r 2 is the solid angle element centered at the direction n̂ inc and
formed by the detector surface at the distance r from the particle.
1
2
∆S ε1 µ
inc 2
0 |E 0 |
The term
on the right-hand side of Eq. (2.97) is proportional to the detector
area ∆S and is equal to the electromagnetic power that would be received by detector
1 in the absence of the scattering particle, whereas − (2π k1 ) ε1 µ 0 Im[E1sca (nˆ inc )
∗
⋅ E inc
0 ] is an attenuation term independent of ∆S , caused by interposing the particle
between the light source and the detector. Thus, a well-collimated detector located in
the far-field zone and having its surface ∆S aligned normal to the exact forwardscattering direction (i.e., nˆ sca = nˆ inc , detector 1 in Fig. 2.3) measures the power of the
incident light attenuated by interference of the incident and the scattered fields plus a
relatively small contribution from the scattered light, whereas a detector with surface
aligned normal to any other scattering direction (i.e., nˆ sca ≠ nˆ inc , detector 2 in Fig.
2.3) “sees” only the scattered light. These are two fundamental features of electromagnetic scattering by a small particle. Equation (2.97) is a representation of the socalled optical theorem and will be further discussed in Section 2.8.
2.6
Phase matrix
In the thought experiment described in the previous section and shown schematically
in Fig. 2.3, it is assumed that the detectors can measure only the total electromagnetic
power and that they make no distinction between electromagnetic waves with different states of polarization. Many detectors of electromagnetic energy are indeed polarization-insensitive. However, by interposing a polarizer between the source of
light and the scattering particle one can generate incident light with a specific state of
polarization, whereas interposing a polarizer between the scattering particle and the
detector enables the detector to measure the power corresponding to a particular polarization component of the scattered light. By repeating the measurement for a num-
50
Scattering, absorption, and emission by small particles
ber of different combinations of the polarizers one can, in principle, determine the
specific prescription for the transformation of a complete set of polarization characteristics of the incident light into that of the scattered light, provided that both sets of
characteristics have the same dimension of energy flux (Section 8.1). As we saw in
Chapter 1, convenient complete sets of polarization characteristics having the dimension of monochromatic energy flux are the coherency and the Stokes vectors. So we
will now assume that the device shown schematically in Fig. 2.3 can (i) generate incident light with different (but physically realizable) combinations of coherency or
Stokes vector components, and (ii) measure the electromagnetic power associated
with any component of the coherency vector or the Stokes vector and equal to the
integral of the component over the surface ∆S of the collimated detector aligned
normal to the direction of propagation r̂. The component itself is then found by dividing the measured power by ∆S .
To derive the relationship between the polarization characteristics of the incident
and the scattered waves for scattering directions away from the incidence direction
(rˆ ≠ nˆ inc ), we first define the respective coherency vectors (cf. Eqs. (1.53), (2.24),
and (2.25)):
Jinc =
1
2
ε1
µ0
Jsca (r nˆ sca ) =
=
inc∗
é E0inc
ϑ E0ϑ ù
ê inc inc∗ ú
ê E0ϑ E0ϕ ú
ê E inc E inc∗ ú,
ê 0ϕ 0ϑ ú
inc∗
êë E0inc
ϕ E0ϕ ú
û
1
2
(2.98)
é Eϑsca (r nˆ sca )[ Eϑsca (r nˆ sca )]∗ ù
ê sca
sca
sca
sca ∗ ú
ε1 ê Eϑ (r nˆ )[ Eϕ (r nˆ )] ú
µ 0 ê Eϕsca (r nˆ sca )[ Eϑsca (r nˆ sca )]∗ ú
ú
ê
êë Eϕsca (r nˆ sca )[ Eϕsca (r nˆ sca )]∗ úû
1 1
r2 2
ˆ sca )[ E1sca
ˆ sca )]∗ ù
é E1sca
ϑ (n
ϑ (n
ê sca sca
sca
sca ∗ ú
ε1 ê E1ϑ (nˆ )[ E1ϕ (nˆ )] ú
.
µ 0 ê E1sca
(nˆ sca )[ E1sca
(nˆ sca )]∗ ú
ϕ
ϑ
ú
ê
sca
sca ˆ sca ∗ ú
êë E1sca
ˆ
(
)[
(
)]
n
E
n
1ϕ
ϕ
û
(2.99)
Equation (2.30) and simple algebra lead to the following formula describing the transformation of the coherency vector of the incident wave into that of the scattered wave:
Jsca (r nˆ sca ) =
1 J sca inc inc
Z (nˆ , nˆ )J ,
r2
(2.100)
where the elements of the coherency phase matrix Z J (nˆ sca , nˆ inc ) are quadratic combinations of the elements of the amplitude scattering matrix S(nˆ sca , nˆ inc ):
2 Scattering, absorption, and emission by an arbitrary finite particle
é |S11|2
ê
*
S11 S 21
J
Z = êê
S S*
ê 21 11
êë |S 21|2
*
S11 S12
*
S12 S11
*
S11 S 22
*
S12 S 21
*
S 21 S12
*
S 22 S11
*
S 21 S 22
*
S 22 S 21
|S12| 2 ù
* ú
S12 S 22
ú.
* ú
S 22 S12
ú
|S 22| 2 úû
51
(2.101)
Analogously, the Stokes phase matrix Z describes the transformation of the Stokes
vector of the incident wave into that of the scattered wave,
Isca (r nˆ sca ) =
1
Z(nˆ sca , nˆ inc )Iinc ,
r2
(2.102)
and is given by
Z(nˆ sca , nˆ inc ) = DZ J (nˆ sca , nˆ inc )D −1,
(2.103)
where
Iinc = DJinc =
1
2
ε1
µ0
inc∗
inc inc∗ ù
é E0inc
ϑ E 0ϑ + E 0ϕ E 0ϕ
ú
ê inc inc∗
inc inc∗
ê E0ϑ E0ϑ − E0ϕ E0ϕ ú
ê − E inc E inc∗ − E inc E inc∗ ú
0ϑ 0ϕ
0ϕ 0ϑ ú
ê
inc∗
inc inc∗ ú
êi( E0inc
ϕ E 0ϑ − E 0ϑ E 0ϕ ) û
ë
(2.104)
and
Isca (r nˆ sca ) = DJsca (r nˆ sca ) =
1 1
r2 2
ε1
µ0
sca ∗
sca sca ∗ ù
é E1sca
ϑ E1ϑ + E1ϕ E1ϕ
ê sca sca ∗
sca sca ∗ ú
ê E1ϑ E1ϑ − E1ϕ E1ϕ ú
ê − E sca E sca ∗ − E sca E sca ∗ ú
1ϕ
1ϑ ú
ê 1ϑ 1ϕ
sca
sca
∗
sca
sca
êëi( E1ϕ E1ϑ − E1ϑ E1ϕ ∗ )úû
(2.105)
(cf. Eq. (1.54)); the matrices D and D−1 were defined by Eqs. (1.55) and (1.57), respectively. Explicit formulas for the elements of the Stokes phase matrix in terms of
the amplitude scattering matrix elements follow from Eqs. (2.101) and (2.103):
Z11 =
1
2
(|S11 |2 + |S12 | 2 + |S 21 |2 + |S 22 | 2 ),
(2.106)
Z12 =
1
2
(|S11 |2 − |S12 |2 + |S 21 |2 − |S 22 | 2 ),
(2.107)
*
*
Z13 = − Re( S11 S12
+ S 22 S 21
),
(2.108)
*
*
Z14 = − Im(S11 S12
− S 22 S 21
),
(2.109)
Z 21 =
1
2
(|S11 |2 + |S12 |2 − |S 21 |2 − |S 22 |2 ),
(2.110)
Z 22 =
1
2
(|S11 |2 − |S12 |2 − |S 21 |2 + |S 22 |2 ),
(2.111)
*
*
Z 23 = − Re( S11 S12
− S 22 S 21
),
(2.112)
*
*
Z 24 = − Im(S11 S12
+ S 22 S 21
),
(2.113)
*
*
Z 31 = − Re( S11 S 21
+ S 22 S12
),
(2.114)
52
Scattering, absorption, and emission by small particles
*
*
Z 32 = − Re( S11 S 21
− S 22 S12
),
(2.115)
*
*
Z 33 = Re( S11 S 22
+ S12 S 21
),
(2.116)
*
*
Z 34 = Im(S11 S 22
+ S 21 S12
),
(2.117)
*
*
Z 41 = − Im(S 21 S11
+ S 22 S12
),
(2.118)
*
*
Z 42 = − Im(S 21 S11
− S 22 S12
),
(2.119)
*
*
Z 43 = Im(S 22 S11
− S12 S 21
),
(2.120)
*
*
).
Z 44 = Re( S 22 S11
− S12 S 21
(2.121)
Finally, the modified Stokes and circular-polarization phase matrices are given by
Z MS (nˆ sca , nˆ inc ) = BZ(nˆ sca , nˆ inc )B −1
(2.122)
Z CP (nˆ sca , nˆ inc ) = AZ(nˆ sca , nˆ inc ) A −1,
(2.123)
and
respectively (see Eqs. (1.59)–(1.66)). The elements of all phase matrices have the
same dimension of area. The matrices Z and Z MS are real-valued. Like the amplitude scattering matrix, the phase matrices explicitly depend on ϕ inc and ϕ sca even
when the incident and/or scattered light propagates along the z-axis.
Up to this point we have considered the scattering of only monochromatic plane
waves. However, it is obvious that Eqs. (2.100) and (2.102) remain valid even when
the incident radiation is a parallel quasi-monochromatic beam of light, provided that
the coherency and Stokes vectors entering these equations are averages over a time
interval long compared with the period of fluctuations (Section 1.6). Hence, the
phase matrix concept is quite useful even in the more general situations involving
quasi-monochromatic light.
In general, all 16 elements of any of the phase matrices introduced above are nonzero. However, the phase matrix elements of a single particle are expressed in terms
of only seven independent real numbers resulting from the four moduli |S ij |
(i, j = 1,2) and three differences in phase between S ij . Therefore, only seven of the
phase matrix elements are actually independent, and there must be nine unique relations among the 16 phase matrix elements. Furthermore, the specific mathematical
structure of the phase matrix can also be used to derive many useful linear and quadratic inequalities for the phase matrix elements. Two important inequalities are
Z11 ≥ 0 (this property follows directly from Eq. (2.106)) and |Z ij | ≤ Z11 (i, j = 1, …,
4). The reader is referred to Hovenier et al. (1986), Cloude and Pottier (1996), and
Hovenier and van der Mee (1996, 2000) for a review of this subject and a discussion
of how the general properties of the phase matrix can be used for testing the results of
theoretical computations and laboratory measurements.
From Eqs. (2.106)–(2.121) and (2.64) we derive the reciprocity relation for the
Stokes phase matrix:
2 Scattering, absorption, and emission by an arbitrary finite particle
53
Z(−nˆ inc , − nˆ sca ) = ∆ 3 [Z(nˆ sca , nˆ inc )]T ∆ 3,
(2.124)
é1
ê
0
T
−1
∆3 = ∆3 = ∆3 = ê
ê0
ê
êë0
(2.125)
where
0ù
ú
1 0 0ú
0 − 1 0ú
ú
0 0 1úû
0
0
and T denotes the transpose of a matrix, as before. The reciprocity relations for other
phase matrices can be obtained easily from Eqs. (2.103), (2.122), and (2.123):
Z J (−nˆ inc , − nˆ sca ) = D −1Z(−nˆ inc , − nˆ sca )D
= D −1 ∆ 3 [Z(nˆ sca , nˆ inc )]T ∆ 3D
= D −1 ∆ 3 [DZ J (nˆ sca , nˆ inc )D −1 ]T ∆ 3D
= D −1 ∆ 3 [D −1 ]T [Z J (nˆ sca , nˆ inc )]T D T ∆ 3D
= ∆ 23 [Z J (nˆ sca , nˆ inc )]T ∆ 23,
(2.126)
Z MS (−nˆ inc , − nˆ sca ) = BZ(−nˆ inc , − nˆ sca )B −1
= B∆ 3 [Z(nˆ sca , nˆ inc )]T ∆ 3B −1
= B∆ 3 [B −1Z MS (nˆ sca , nˆ inc )B]T ∆ 3B −1
= B∆ 3B T [Z MS (nˆ sca , nˆ inc )]T [B −1 ]T ∆ 3B −1
= ∆ MS [Z MS (nˆ sca , nˆ inc )]T [ ∆ MS ]−1,
(2.127)
Z CP (−nˆ inc , − nˆ sca ) = A∆ 3 A T [Z CP (nˆ sca , nˆ inc )]T [ A −1 ]T ∆ 3 A −1
= [Z CP (nˆ sca , nˆ inc )]T ,
(2.128)
0
é1 0
ê
0 −1 0
=ê
ê0 0 − 1
ê
0
êë0 0
0ù
ú
0ú
,
0ú
ú
1úû
(2.129)
0
é1 2 0
ê
0 12 0
= [ ∆ MS ]T = ê
ê0
0 −1
ê
0
0
êë 0
0ù
ú
0ú
,
0ú
ú
1úû
where
−1
∆ 23 = ∆ T23 = ∆ 23
∆ MS
é2
ê
0
[ ∆ MS ]−1 = ê
ê0
ê
êë0
0ù
ú
2 0 0ú
. (2.130)
0 − 1 0ú
ú
0 0 1úû
0
0
The backscattering theorem, Eq. (2.65), along with Eqs. (2.106), (2.111), (2.116), and
(2.121), leads to the following general property of the backscattering Stokes phase
54
Scattering, absorption, and emission by small particles
matrix (Mishchenko et al. 2000b):
Z11(−nˆ , nˆ ) − Z 22 (−nˆ , nˆ ) + Z 33(−nˆ , nˆ ) − Z 44 (−nˆ , nˆ ) = 0.
(2.131)
Electromagnetic scattering most typically produces light with polarization characteristics different from those of the incident beam. If the incident beam is unpolarized, i.e., Iinc = [ I inc 0 0 0]T , the scattered light generally has at least one non-zero
Stokes parameter other than intensity:
I sca = Z11 I inc ,
Q sca = Z 21 I inc ,
U sca = Z 31 I inc ,
V sca = Z 41 I inc.
(2.132)
This phenomenon is traditionally called “polarization” and results in scattered light
with non-zero degree of polarization, see Eq. (1.111),
P=
2
2
2
+ Z 31
+ Z 41
Z 21
.
Z11
(2.133)
Obviously, if the incident light is unpolarized, then the element Z11 determines the
angular distribution of the scattered intensity. When the incident beam is linearly
polarized, i.e., Iinc = [ I inc Q inc U inc 0]T , the scattered light may become elliptically
polarized (V sca ≠ 0). Conversely, when the incident light is circularly polarized, i.e.,
Iinc = [ I inc 0 0 V inc ]T , the scattered light may become partially linearly polarized
(Q sca ≠ 0 and/or U sca ≠ 0). A general feature of scattering by a single particle is that
if the incident beam is fully polarized ( P inc = 1) then the scattered light is also fully
polarized. Hovenier et al. (1986) gave a proof of this property based on the general
mathematical structure of the Stokes phase matrix. Thus, a single particle does not
depolarize fully polarized incident light. We will see later, however, that single scattering by a collection of non-identical nonspherical particles (including particles of
the same kind but with different orientations) can result in depolarization of the incident polarized light, and this is another important property of electromagnetic scattering.
2.7
Extinction matrix
Let us now consider the special case of the exact forward-scattering direction
(rˆ = nˆ inc ). As in Section 2.6, we begin by defining the coherency vector of the total
field for r̂ very close to n̂ inc as
J(r rˆ ) =
1
2
é Eϑ (r rˆ )[ Eϑ (r rˆ )]∗ ù
ú
ê
ε1 ê Eϑ (r rˆ )[ Eϕ (r rˆ )]∗ ú
,
µ 0 ê Eϕ (r rˆ )[ Eϑ (r rˆ )]∗ ú
ú
ê
ê Eϕ (r rˆ )[ Eϕ (r rˆ )]∗ ú
û
ë
(2.134)
2 Scattering, absorption, and emission by an arbitrary finite particle
55
where the total electric field is
E(r rˆ ) = E inc (r rˆ ) + E sca (rrˆ ).
(2.135)
Integrating the elements of J(r rˆ ) over the surface of the collimated detector aligned
normal to n̂ inc and using Eqs. (2.92), (2.94), and (2.98), we derive after rather lengthy
algebraic manipulations
J(r nˆ inc )∆S = Jinc ∆S − Κ J (nˆ inc )Jinc + O(r −2 ),
(2.136)
where the elements of the 4× 4 so-called coherency extinction matrix Κ J (ϑ inc , ϕ inc )
are expressed in terms of the elements of the forward-scattering amplitude matrix
S(ϑ inc , ϕ inc; ϑ inc , ϕ inc ) as follows:
∗
é S11
− S11
ê
∗
i 2π ê S 21
ΚJ =
k1 ê − S 21
ê
êë 0
∗
S12
− S12
∗
− S11
S 22
0
0
∗
S11
− S 22
∗
S 21
− S 21
ù
ú
− S12 ú
ú.
∗
S12
ú
∗
S 22
− S 22 úû
0
(2.137)
Switching again to Stokes parameters, we have
I(r nˆ inc )∆S = Iinc ∆S − Κ(nˆ inc )Iinc + O(r −2 ),
(2.138)
where I(r nˆ inc ) = DJ(r nˆ inc ). The Stokes extinction matrix is given by
Κ(nˆ inc ) = DΚ J (nˆ inc )D −1 .
(2.139)
The explicit formulas for the elements of this matrix in terms of the elements of the
matrix S(ϑ inc , ϕ inc; ϑ inc , ϕ inc ) are as follows:
Κ jj =
2π
Im(S11 + S 22 ),
k1
Κ 12 = Κ 21 =
2π
Im(S11 − S 22 ),
k1
Κ 13 = Κ 31 = −
Κ 14 = Κ 41 =
j = 1, ..., 4,
2π
Im(S12 + S 21),
k1
2π
Re( S 21 − S12 ),
k1
Κ 23 = −Κ 32 =
2π
Im(S 21 − S12 ),
k1
Κ 24 = −Κ 42 = −
Κ 34 = −Κ 43 =
2π
Re( S12 + S 21),
k1
2π
Re( S 22 − S11).
k1
(2.140)
(2.141)
(2.142)
(2.143)
(2.144)
(2.145)
(2.146)
56
Scattering, absorption, and emission by small particles
Thus, only seven elements of the Stokes extinction matrix are independent. It is easy
to verify that this is also true of the coherency extinction matrix. The elements of
both matrices have the dimension of area and explicitly depend on ϕ inc even when
the incident wave propagates along the z-axis.
Equations (2.136) and (2.138) represent the most general form of the optical theorem. They show that the presence of the scattering particle changes not only the total
power of the electromagnetic radiation received by the detector facing the incident
wave (detector 1 in Fig. 2.3) but also, perhaps, its state of polarization. This phenomenon is called dichroism and results from different attenuation rates for different
polarization components of the incident wave. Obviously, Eqs. (2.136) and (2.138)
remain valid if the incident radiation is a parallel quasi-monochromatic beam of light
rather than a monochromatic plane wave.
From Eqs. (2.64) and (2.140)–(2.146) we obtain the reciprocity relation for the
Stokes extinction matrix:
Κ(−nˆ inc ) = ∆ 3 [Κ(nˆ inc )]T ∆ 3,
(2.147a)
where the matrix ∆ 3 is given by Eq. (2.125). It is also easy to derive a related symmetry property:
é Κ11 (nˆ inc )
Κ12 (nˆ inc ) − Κ13 (nˆ inc ) Κ14 (nˆ inc ) ù
ú
ê
Κ 21 (nˆ inc )
Κ 22 (nˆ inc )
Κ 23 (nˆ inc ) − Κ 24 (nˆ inc )ú
ê
inc
ˆ
.
Κ(−n ) = ê
− Κ 31 (nˆ inc ) Κ 32 (nˆ inc )
Κ 33 (nˆ inc )
Κ 34 (nˆ inc ) ú
ú
ê
êë Κ 41 (nˆ inc ) − Κ 42 (nˆ inc ) Κ 43 (nˆ inc )
Κ 44 (nˆ inc ) úû
(2.147b)
In other words, the only effect of reversing the direction of propagation is to change
the sign of four elements of the Stokes extinction matrix. The modified Stokes and
circular-polarization extinction matrices are given by
Κ MS (nˆ inc ) = BΚ (nˆ inc )B −1,
CP
ˆ inc
ˆ inc
−1
Κ (n ) = AΚ(n ) A .
(2.148)
(2.149)
Reciprocity relations for the matrices Κ J (nˆ inc ), Κ MS (nˆ inc ), and Κ CP (nˆ inc ) can be
derived from Eq. (2.147a) by analogy with Eqs. (2.126)–(2.128):
Κ J (−nˆ inc ) = ∆ 23 [Κ J (nˆ inc )]T ∆ 23,
(2.150)
Κ MS (−nˆ inc ) = ∆ MS [Κ MS (nˆ inc )]T [ ∆ MS ]−1,
(2.151)
CP
ˆ inc
CP
ˆ inc
T
Κ (−n ) = [Κ (n )] .
2.8
(2.152)
Extinction, scattering, and absorption cross sections
Knowledge of the total electromagnetic field in the far-field zone also allows us to
calculate such important optical characteristics of the scattering object as the total
2 Scattering, absorption, and emission by an arbitrary finite particle
57
scattering, absorption, and extinction cross sections. These optical cross sections are
defined as follows. The product of the scattering cross section Csca and the incident
monochromatic energy flux gives the total monochromatic power removed from the
incident wave as a result of scattering of the incident radiation in all directions.
Analogously, the product of the absorption cross section Cabs and the incident monochromatic energy flux gives the total monochromatic power removed from the incident wave as a result of absorption of light by the object. Of course, the absorbed
electromagnetic energy does not disappear but, rather, is converted into other forms of
energy. Finally, the extinction cross section Cext is the sum of the scattering and absorption cross sections and, when multiplied by the incident monochromatic energy
flux, gives the total monochromatic power removed from the incident light by the
combined effect of scattering and absorption.
To determine the total optical cross sections, we surround the object by an imaginary sphere of radius r large enough to be in the far-field zone. Since the surrounding
medium is assumed to be nonabsorbing, the net rate at which the electromagnetic
energy crosses the surface S of the sphere is always non-negative and is equal to the
power absorbed by the particle:
W abs = −
dS áS(r )ñ ⋅ rˆ = −r 2
drˆ áS(r )ñ ⋅ rˆ
(2.153)
4π
S
(see Eq. (1.31)). According to Eq. (2.88), W abs can be written as a combination of
three terms:
W abs = W inc − W sca + W ext ,
(2.154)
where
W inc = −r 2
drˆ áS inc (r )ñ ⋅ rˆ ,
4π
W ext = −r 2
W sca = r 2
drˆ áS sca (r )ñ ⋅ rˆ ,
4π
drˆ áS ext (r )ñ ⋅ rˆ.
(2.155)
4π
W inc vanishes identically because the surrounding medium is nonabsorbing and
S inc (r ) is a constant vector independent of r, whereas W sca is the rate at which the
scattered energy crosses the surface S in the outward direction. Therefore, W ext is
equal to the sum of the energy scattering rate and the energy absorption rate:
W ext = W sca + W abs .
(2.156)
Inserting Eqs. (2.90)–(2.95) in Eq. (2.155) and recalling the definitions of the extinction and scattering cross sections, we derive after some algebra
Cext =
W ext
1
2
ε1 µ
inc 2
0 |E 0 |
=
4π
∗
Im[ E1sca (nˆ inc ) ⋅ E inc
0 ],
inc 2
k1 |E 0 |
(2.157)
Scattering, absorption, and emission by small particles
58
Csca =
W sca
ε1 µ
1
2
inc 2
0 |E 0 |
=
1
2
|E inc
0 |
drˆ |E1sca (rˆ )|2.
(2.158)
4π
In view of Eqs. (2.24), (2.30), (2.102), (2.104), (2.105), and (2.140)–(2.146), Eqs.
(2.157) and (2.158) can be rewritten as
Cext =
1
I
inc
[Κ 11(nˆ inc ) I inc + Κ 12 (nˆ inc )Q inc + Κ 13(nˆ inc )U inc + Κ 14 (nˆ inc )V inc ],
Csca =
=
r2
I inc
drˆ I sca (r rˆ )
4π
1
I
(2.159)
drˆ [ Z11(rˆ , nˆ inc ) I inc + Z12 (rˆ , nˆ inc )Q inc
inc
4π
+ Z13(rˆ , nˆ inc )U inc + Z14 (rˆ , nˆ inc )V inc ].
(2.160)
The absorption cross section is equal to the difference of the extinction and scattering
cross sections:
Cabs = Cext − Csca ≥ 0.
(2.161)
The single-scattering albedo is defined as the ratio of the scattering and extinction
cross sections:
ϖ=
Csca
≤ 1.
Cext
(2.162)
This quantity is widely used in radiative transfer theory and is interpreted as the probability that a photon interacting with the particle will be scattered rather than absorbed. Obviously, ϖ = 1 for nonabsorbing particles. Equations (2.159) and (2.160)
(and thus Eqs. (2.161) and (2.162)) also hold for quasi-monochromatic incident light
provided that the elements of the Stokes vector entering these equations are averages
over a time interval long compared with the period of fluctuations. All cross sections
are inherently real-valued positive quantities and have the dimension of area. They
depend on the direction, polarization state, and wavelength of the incident light as
well as on the particle size, morphology, relative refractive index, and orientation
with respect to the reference frame.
Equation (2.159) is another representation of the optical theorem and, along with
Eqs. (2.140)–(2.143), shows that although extinction is the combined effect of absorption and scattering in all directions by the particle, it is determined only by the
amplitude scattering matrix in the exact forward direction. This is a direct consequence of the fact that extinction results from interference between the incident and
scattered light, Eq. (2.91), and the presence of delta-function terms in Eqs. (2.92) and
(2.93). Having derived Eq. (2.157), we can now rewrite Eq. (2.97) in the form
W∆S (nˆ inc ) =
1
2
ε1 inc 2
|E 0 | (∆S − Cext ) + O(r − 2 ).
µ0
(2.163)
2 Scattering, absorption, and emission by an arbitrary finite particle
59
This shows that the extinction cross section is a well-defined, observable quantity and
can be determined by measuring W∆S (nˆ inc ) with and without the particle interposed
between the source of light and the detector. The net effect of the particle is to reduce
the detector area by “casting a shadow” of area Cext . Of course, this does not mean
that Cext is merely given by the area G of the particle geometrical projection on the
detector surface. However, this geometrical interpretation of the extinction cross section illustrates the rationale for introducing the dimensionless efficiency factor for
extinction as the ratio of the extinction cross section to the geometrical cross section:
Qext =
Cext
.
G
(2.164)
We will see in later chapters that Qext can be considerably greater or much less than
unity. The efficiency factors for scattering and absorption are defined analogously:
Qsca =
Csca
,
G
Qabs =
Cabs
.
G
(2.165)
The quantity
dCsca I sca (r rˆ )r 2
=
dΩ
I inc
=
1
I
inc
[ Z11(rˆ , nˆ inc ) I inc + Z12 (rˆ , nˆ inc )Q inc + Z13(rˆ , nˆ inc )U inc + Z14 (rˆ , nˆ inc )V inc ]
(2.166)
also has the dimension of area and is called the differential scattering cross section; it
describes the angular distribution of the scattered light and specifies the electromagnetic power scattered into unit solid angle about a given direction per unit incident
intensity. (Note that the symbol dCsca dΩ should not be interpreted as the derivative
of a function of Ω .) The differential scattering cross section depends on the polarization state of the incident light as well as on the incidence and scattering directions.
Clearly,
drˆ
Csca =
4π
dCsca
dΩ
(cf. Eqs. (2.160) and (2.166)). A quantity related to the differential scattering cross
section is the phase function p (rˆ , nˆ inc ) defined as
p (rˆ , nˆ inc ) =
4π dCsca
.
Csca dΩ
(2.167)
The convenience of the phase function is that it is dimensionless and normalized:
1
4π
drˆ p (rˆ , nˆ inc ) = 1.
4π
(2.168)
60
Scattering, absorption, and emission by small particles
The asymmetry parameter ácos Θ ñ is defined as the average cosine of the scattering
angle Θ = arccos (rˆ ⋅ nˆ inc ) (i.e., the angle between the incidence and scattering directions):
ácosΘ ñ =
1
4π
drˆ p (rˆ , nˆ inc )rˆ ⋅ nˆ inc =
4π
1
Csca
drˆ
4π
dCsca
rˆ ⋅ nˆ inc.
dΩ
(2.169)
The asymmetry parameter is positive if the particle scatters more light toward the
forward direction (Θ = 0), is negative if more light is scattered toward the backscattering direction (Θ = π ), and vanishes if the scattering is symmetric with respect to
the plane perpendicular to the incidence direction. Obviously, á cosΘ ñ ∈ [−1, + 1].
The limiting values correspond to the phase functions 4π δ(rˆ + nˆ inc ) and 4π δ(r̂ −
nˆ inc ), respectively.
2.9
Radiation pressure and radiation torque
The scattering and absorption of an electromagnetic wave cause the transfer of momentum from the electromagnetic field to the scattering object. The resulting force,
called radiation pressure, is used in laboratories to levitate and size small particles
(Ashkin and Dziedzic 1980; Chýlek et al. 1992; Ashkin 2000) and affects the spatial
distribution of interplanetary and interstellar dust grains (Il’in and Voshchinnikov
1998; Landgraf et al. 1999). If the amplitudes of the incident and scattered fields do
not change in time, the force due to radiation pressure averaged over the period 2π ω
of the time-harmonic incident wave is
t
dS áTM (r )ñ ⋅ nˆ
F=
(2.170)
S
t
(Stratton 1941, Section 2.5; Jackson 1998, Section 6.7), where TM is the so-called
Maxwell stress tensor, the integration is performed over a closed surface S surrounding the scattering object, and n̂ is the unit vector in the direction of the local outward
normal to S. Assume, for simplicity, that the scattering object is surrounded by a vacuum. Then the instantaneous value of the Maxwell stress tensor is
t
t
TM = ε 0 [E ⊗ E + c 2 B ⊗ B − 12 (E ⋅ E + c 2 B ⋅ B) I ]
t
= ε 0 E ⊗ E + µ 0 H ⊗ H − 12 (ε 0 E ⋅ E + µ 0 H ⋅ H ) I ,
(2.171)
where c = 1
ε 0 µ 0 is the speed of light in a vacuum. By analogy with Eq. (1.24),
we have for the time average of the Maxwell stress tensor
t
t
áTM (r )ñ = 12 Re[ε 0 E(r ) ⊗ E∗(r ) + µ 0 H (r ) ⊗ H ∗(r ) − 12 (ε 0 |E(r )|2 + µ 0 |H (r )|2 ) I ].
(2.172)
2 Scattering, absorption, and emission by an arbitrary finite particle
61
It is convenient to choose for S a sphere centered at the scattering object and having a
radius r large enough to be in the far-field zone. Then Eq. (2.170) becomes
t
drˆ áTM (r rˆ )ñ ⋅ rˆ.
F = r2
(2.173)
4π
The total electric and magnetic fields are vector sums of the respective incident and
scattered fields given by Eqs. (2.92)–(2.95). Because the incident and scattered fields
are transverse, the first and second terms in square brackets on the right-hand side of
Eq. (2.172) do not contribute to the integral in Eq. (2.173). We thus have
F=−
ε 0r 2
Re
4
drˆ rˆ{|E inc (r )| 2 + |E sca (r )|2 +E inc (r ) ⋅ [E sca (r )]∗
4π
+ E sca (r ) ⋅ [E inc (r )]∗ }
−
µ0r 2
Re
4
drˆ rˆ{|H inc (r )| 2 + |H sca (r )|2 + H inc (r ) ⋅ [H sca (r )]∗
4π
+ H sca (r ) ⋅ [H inc (r )]∗ }.
(2.174)
The terms |E inc (r )|2 and |H inc (r )|2 are constants independent of r, so their contribution to F is simply zero. The contribution of the remaining terms follows from Eqs.
(2.92)–(2.95) and the vector identity (a × b) ⋅ (c × d) = (a ⋅ c)(b ⋅ d) − (a ⋅ d)(b ⋅ c):
F=
2πε 0 inc
ε0
∗
nˆ Im{E1sca (nˆ inc ) ⋅ E inc
0 }−
2
k1
drˆ rˆ |E1sca (rˆ )| 2
(2.175)
4π
or, in view of Eqs. (2.102), (2.104), (2.105), (2.157), and (2.166),
1
1
F = nˆ inc Cext I inc −
c
c
drˆ rˆ [ Z11(rˆ , nˆ inc ) I inc + Z12 (rˆ , nˆ inc )Q inc
4π
+ Z13(rˆ , nˆ inc )U inc + Z14 (rˆ , nˆ inc )V inc ]
1
1
= nˆ inc Cext I inc − I inc
c
c
drˆ rˆ
4π
dCsca
dΩ
(2.176)
(Mishchenko 2001).
Although the first term on the right-hand side of Eq. (2.176) represents a force in
the direction of nˆ inc , the direction of the total radiation force is different, in general,
from the direction of propagation of the incident beam and depends on its polarization
state because of the second term. The projection of the total force on any direction n̂
is simply the dot product F ⋅ n̂. In particular, the component of the force in the direction of propagation of the incident light is
1
1
F ⋅ nˆ inc = Cext I inc − I inc
c
c
=
drˆ rˆ ⋅ nˆ inc
4π
1 inc
I (Cext − Csca ácosΘ ñ )
c
dCsca
dΩ
62
Scattering, absorption, and emission by small particles
=
1 inc
I C pr
c
(2.177)
(see Eq. (2.169)), where the quantity
Cpr = Cext − Csca ácos Θ ñ
(2.178)
is called the radiation-pressure cross section. By analogy with Eqs. (2.164) and
(2.165), we can define the radiation-pressure efficiency factor as
Qpr =
Cpr
.
G
(2.179)
Although being the result of a lengthy rigorous derivation, Eq. (2.177) allows a
transparent physical interpretation. A beam of light carries linear momentum as well
as energy. The direction of the momentum is that of propagation, while the absolute
value of the momentum is energy/(speed of light). Since the total momentum of the
electromagnetic field and the scattering object must be constant, the radiation force
exerted on the object is equal to the momentum removed from the total electromagnetic field per unit time. Consider the component of the force in the direction of incidence. The momentum removed from the incident beam per unit time is Cext I inc c.
Of this amount, the part proportional to Cabs is not replaced, whereas the part proportional to Csca is to some extent replaced by the contribution due to the projection of
the moment of the scattered light on the direction of incidence. This contribution is
equal to the integral of I sca cosΘ c over all scattering directions, or
I inc Csca ácosΘ ñ c. Note that van de Hulst (1957) used similar arguments as an heuristic derivation of Eq. (2.177).
If the absolute temperature of the particle is above zero then light emitted by the
particle in all directions causes an additional component of the radiation force. This
component will be discussed in Section 2.10.
The radiation pressure is accompanied by the radiation torque exerted on the particle and given by
t
dS r rˆ ⋅ [áTM (r )ñ × rˆ ]
Γ=−
S
t
drˆ rˆ ⋅ [áTM (r )ñ × rˆ ]
= −r 3
(2.180)
4π
(cf. p. 288 of Jackson 1998), where r is the radius of a sphere S centered inside the
t
scattering particle and having its surface in the far-field zone. Since rˆ ⋅ I × rˆ vanishes
identically, only the first two terms in square brackets on the right-hand side of Eq.
(2.172) contribute to the integrals in Eq. (2.180). The evaluation of this contribution
is complicated because it requires the knowledge of not only the transverse component of the scattered electric and magnetic fields but also of the longitudinal component, which we have so far neglected because it decays faster than 1 r. Marston and
2 Scattering, absorption, and emission by an arbitrary finite particle
63
Crichton (1984) computed Γ for homogeneous and isotropic spherical particles,
whereas Draine and Weingartner (1996) derived a formula for Γ in the framework of
the so-called discrete dipole approximation (see Section 6.5).
2.10 Thermal emission
If the particle’s absolute temperature T is above zero, it can emit as well as scatter and
absorb electromagnetic radiation. The emitted radiation in the far-field zone of the
particle propagates in the radial direction, i.e., along the unit vector rˆ = r r, where r
is the position vector of the observation point with origin inside the particle. The energetic and polarization characteristics of the emitted radiation are described by a
four-component Stokes emission column vector Κ e (rˆ , T , ω ) defined in such a way
that the net rate at which the emitted energy crosses a surface element ∆S normal to
r̂ at a distance r from the particle at frequencies from ω to ω + ∆ω is
We =
1
Κ e1(rˆ , T , ω )∆S∆ω .
r2
(2.181)
Κ e1(rˆ , T , ω ), the first component of the column vector, can also be interpreted as the
amount of electromagnetic energy emitted by the particle in the direction r̂ per unit
solid angle per unit frequency interval per unit time.
In order to calculate Κ e (rˆ , T , ω ), let us assume that the particle is placed inside an
opaque cavity of dimensions large compared with the particle and any wavelength
under consideration (Fig. 2.4a). If the cavity and the particle are maintained at the
constant absolute temperature T, then the equilibrium electromagnetic radiation inside
the cavity is isotropic, homogeneous, and unpolarized (Mandel and Wolf 1995). This
radiation can be represented as a collection of quasi-monochromatic, unpolarized,
incoherent beams propagating in all directions and characterized by the Planck blackbody energy distribution I b (T , ω ). Specifically, at any point inside the cavity the
amount of radiant energy per unit frequency interval, confined to a small solid angle
∆Ω about any direction, which crosses an area ∆S normal to this direction in unit
time is given by
I b (T , ω )∆S∆Ω =
hω 3
é æ hω ö ù
÷÷ − 1ú
4π c êexpçç
ë è k BT ø û
∆S∆Ω ,
(2.182)
3 2
where h = h 2π , h is Planck’s constant, c is the speed of light in a vacuum, and k B is
Boltzmann’s constant.
Consider an imaginary collimated, polarization-sensitive detector of electromagnetic radiation with surface ∆S and small solid-angle field of view ∆Ω , placed at a
distance r from the particle (Fig. 2.4a). The dimension of the detector surface is much
greater than any dimension of the particle and r is large enough to be in the far-field
64
Scattering, absorption, and emission by small particles
S
n̂
tor
Detec
Particle
(a)
Ω
Ω
Particle
(b)
Figure 2.4. (a) Cavity, particle, and electromagnetic radiation field in thermal equilibrium.
(b) Illumination geometry.
zone of the particle but smaller than
∆S ∆Ω . The latter condition ensures that all
plane wave fronts incident on the detector in directions falling into its solid-angle
field of view ∆Ω are equally attenuated by the particle (Fig. 2.4b). The surface ∆S
is aligned normal to and centered on r̂, where r̂ is the unit vector originating inside
the particle and pointing toward the detector.
In the absence of the particle, the polarized signal per unit frequency interval
measured by the detector would be given by
Ib (T , ω )∆S∆Ω ,
where
(2.183)
2 Scattering, absorption, and emission by an arbitrary finite particle
é I b (T , ω )ù
ê
ú
0 ú
I b (T , ω ) = ê
ê 0 ú
ê
ú
êë 0 úû
65
(2.184)
is the blackbody Stokes column vector. The particle attenuates the incident blackbody radiation, emits radiation, and scatters the blackbody radiation coming from all
directions in the direction of the detector. Taking into account that only the radiation
emitted and scattered by the particle within the solid-angle field of view ∆Ω is detected (Fig. 2.4b), we conclude that the polarized signal measured by the detector in
the presence of the particle is
drˆ ′ Z (rˆ , rˆ ′, ω )Ib (T , ω )
I b (T , ω )∆S∆Ω − Κ(rˆ , ω )I b (T , ω )∆Ω + Κ e (rˆ , T , ω )∆Ω + ∆Ω
4π
(2.185)
(see Eqs. (2.138) and (2.102)). However, in thermal equilibrium the presence of the
particle does not change the distribution of radiation. Therefore, we can equate expressions (2.183) and (2.185) and finally derive for the ith component of Κ e
drˆ ′ Z i1(rˆ , rˆ ′, ω ),
Κ ei (rˆ , T , ω ) = I b (T , ω )Κ i1(rˆ , ω ) − I b (T , ω )
i = 1, ..., 4.
(2.186)
4π
This important relation expresses the Stokes emission vector in terms of the leftmost
columns of the extinction and phase matrices and the Planck energy distribution.
Although our derivation assumed that the particle was in thermal equilibrium with the
surrounding radiation field, emissivity is a property of the particle only. Therefore,
Eq. (2.186) is valid for any particle, in equilibrium or in nonequilibrium. A more
detailed derivation of this formula based on the so-called fluctuation-dissipation theorem is given by Tsang et al. (2000).
As we pointed out in Section 2.9, the emitted radiation contributes to the total
radiation force exerted on the particle. The emitted radiation is incoherent and does
not interact with the incident and scattered radiation, thereby generating an independent component of the radiation force. Furthermore, emission is analogous to scattering in that it generates radiation propagating radially in all directions. Therefore, we
can write the emission component of the radiation force by analogy with the scattering component given by the second term on the right-hand side of Eq. (2.176):
Fe (T ) = −
1
c
∞
dω
0
drˆ rˆΚ e1(rˆ , T , ω ).
(2.187)
4π
Unlike the extinction and scattering components of the radiation force, the emission
component depends on the particle temperature. Another effect of emission is to pro-
66
Scattering, absorption, and emission by small particles
Observation
point
r1
1
r2
r12 Scattering
object
2
Figure 2.5. The amplitude matrix changes when the origin is shifted. The position vectors r1
and r2 originate at points 1 and 2, respectively.
duce a component of the radiation torque independent of that caused by scattering and
absorption.
2.11 Translations of the origin
We began Section 2.2 by choosing the origin of the coordinate system close to the
geometrical center of the scattering object, and that step was essential in deriving the
formulas describing electromagnetic scattering in the far-field zone. Although the
origin can be chosen arbitrarily, in general, the amplitude scattering matrix will
change if the origin is moved, even if the orientation of the particle with respect to the
reference frame remains the same. It is, therefore, important to supplement the rotation transformation law of Eq. (2.72) by a translation law describing the transformation of the amplitude scattering matrix upon a shift in the origin.
Let us consider two reference frames with origins 1 and 2 inside a scattering particle (Fig. 2.5). We assume that both reference frames have the same spatial orientation and denote the respective amplitude scattering matrices as S1 and S 2. Let the
particle be illuminated by a plane electromagnetic wave
ˆ inc ⋅ r1),
E inc (r1) = E inc
0 exp(ik1n
(2.188)
where r1 is the position vector originating at origin 1. Let r2 be the position vector
of the same observation point but originating at origin 2. Since r1 = r2 + r12 , where
r12 connects origins 1 and 2, the incident electric field at the same observation point
can also be written as
ˆ inc ⋅ r2 ) exp(ik1nˆ inc ⋅ r12 ).
E inc (r2 ) = E inc
0 exp(ik1n
(2.189)
The scattered field at an observation point does not depend on how we choose the
origin of the coordinate system as long as the incident field remains the same. Therefore, we have in the far-field zone
2 Scattering, absorption, and emission by an arbitrary finite particle
67
inc
inc
ær
ö é E0ϑ ù e ik r
ær
ö é E0ϑ ù
e ik r
S1 çç 1 , nˆ inc ÷÷ ê inc ú =
exp(ik1nˆ inc ⋅ r12 ) S 2 çç 2 , nˆ inc ÷÷ ê inc ú.
r1
r2
è r1
ø êë E0ϕ úû
è r2
ø êë E0ϕ úû
1 1
1 2
(2.190)
Taking the limits k1r1 → ∞ and k1r2 → ∞ and using the law of cosines,
r22 = r12 + r122 − 2r1 ⋅ r12 ,
we finally obtain
S1(nˆ sca , nˆ inc ) = e i∆ S 2 (nˆ sca , nˆ inc ),
(2.191)
where nˆ sca = r1 r1,
∆ = k1 r12 ⋅ (nˆ inc − nˆ sca ),
(2.192)
and we have assumed that
r12
n1
r1
and
k1 r122
n 1.
2r1
(2.193)
Equation (2.191) is the sought translation transformation law for the amplitude scattering matrix. It remains valid even if either origin lies outside the scattering particle,
as long as the asymptotic far-field conditions of Eq. (2.193) are satisfied.
Despite the fact that the amplitude scattering matrix changes when the origin is
shifted, the extinction and phase matrices remain invariant. Indeed, the factor e i∆ is
common to all elements of the amplitude scattering matrix and disappears when multiplied by its complex-conjugate counterpart, whereas the phase ∆ vanishes identically in the exact forward-scattering direction (cf. Eqs. (2.106)–(2.121) and (2.140)–
(2.146)). It is straightforward to verify that all optical cross sections and efficiency
factors, the single-scattering albedo, the phase function, the asymmetry parameter, the
emission vector, and the radiation force also remain unchanged.
Further reading
The books by Colton and Kress (1983), Varadan et al. (1991) and de Hoop (1995)
provide a thorough theoretical introduction to the propagation and scattering of electromagnetic, acoustic, and elastodynamic waves. Appendix 3 of Van Bladel (1964)
and chapter 6 of Varadan et al. (1991) list important formulas from vector and dyadic
algebra and vector and dyadic calculus. The use of dyadics and dyadic Green’s functions in electromagnetics is described by Tai (1993).
Chapter 3
Scattering, absorption, and emission by collections
of independent particles
The formalism developed in the preceding chapter strictly applies only to the far-field
scattering and absorption of monochromatic or quasi-monochromatic light by an isolated particle in the form of a single body or a fixed finite aggregate (Fig. 2.1) and to
the thermal emission from such a particle. We will now describe how this formalism
can be extended to single and multiple scattering, absorption, and emission by collections of independently scattering particles under certain simplifying assumptions.
3.1
Single scattering, absorption, and emission by a small
volume element comprising randomly and sparsely
distributed particles
Consider first a small volume element having a linear dimension l, comprising a
number N of randomly positioned particles, and illuminated by a plane electromagnetic wave. Although the volume element is assumed to be macroscopically small, its
size must still be much larger than the size of the constituent particles and the wavelength of the incident light. We assume that N is sufficiently small that the mean distance between the particles is also much larger than the incident wavelength and the
average particle size. We also assume that N is sufficiently small that the main contribution to the total scattered radiation exiting the volume element comes from light
scattered only once. In other words, the contribution to the total scattered signal of
light scattered two and more times by particles inside the volume element is assumed
to be negligibly small. This is equivalent to requiring that the “scattering efficiency”
68
3 Scattering by collections of independent particles
69
N áCsca ñl −2 of the volume element (i.e., the ratio of the total scattering cross section
of the particles contained in the volume element to the volume element’s geometrical
cross section) be much smaller than unity; áCsca ñ is the average scattering cross section per particle. Finally, we assume that the positions of the particles during the
measurement are sufficiently random that there are no systematic phase relations between individual waves scattered by different particles.
Let the incident electric field be given by
ˆ inc ⋅ r ),
E inc (r ) = E inc
0 exp(ik1n
(3.1)
where the position vector r originates at the geometrical center of the volume element
O. The total electric field scattered by the volume element at a large distance r from
O can be written as the vector sum of the partial fields scattered by the component
particles:
N
sca
E sca
n (r ),
E (r ) =
(3.2)
n=1
where the position vector r of the observation point originates at O, and the index n
numbers the particles. Since we ignore multiple scattering, we assume that each particle is excited only by the external incident field but not by the secondary fields
scattered by other particles. Furthermore, because the particles are widely separated,
each of them scatters the incident wave in exactly the same way as if all other particles did not exist. Therefore, according to Section 2.11, the partial scattered fields are
given by
é[E sca
é E0inc
e ik r i∆
ϑ ù
n (r )]ϑ ù
e S n (rˆ , nˆ inc ) ê inc ú,
ê sca
ú =
r
ëê[E n (r )]ϕ ûú
ëê E0ϕ ûú
1
n
(3.3)
where rˆ = r r , S n (rˆ , nˆ inc ) is the amplitude scattering matrix of the nth particle, centered inside that particle, the phase ∆ n is given by
∆ n = k1rOn ⋅ (nˆ inc − rˆ ),
(3.4)
and the vector rOn connects the origin of the volume element O with the nth particle’s
origin. As in Section 2.11, we have assumed that
rOn
n 1 and
r
2
k1rOn
2r
n 1,
n = 1, ..., N,
(3.5)
or, equivalently,
l
r
n 1 and
k1l 2
2r
n 1.
(3.6)
These conditions explicitly indicate that the observation point is in the far-field zone
of the small volume element as a whole and that the latter is treated as a single scatterer. We thus have
70
Scattering, Absorption, and Emission of Light by Small Particles
é Eϑsca (r )ù e ik r
é E0inc
ϑ ù
inc
ˆ
ˆ
=
S
(
r
,
n
)
ê sca ú
ê inc ú,
r
êë Eϕ (r )úû
êë E0ϕ úû
1
(3.7)
where the total amplitude scattering matrix of the small volume element centered at O
is
N
e i∆n S n (rˆ , nˆ inc ).
inc
S(rˆ , nˆ ) =
(3.8)
n =1
Since the ∆ n vanish in the exact forward-scattering direction, substituting Eq.
(3.8) in Eqs. (2.140)–(2.146) yields the total extinction matrix of the small volume
element:
N
Κ=
Κ n = N áΚñ,
(3.9)
n =1
where áΚñ is the average extinction matrix per particle. Equation (2.159) then gives
the total extinction cross section:
N
Cext =
(Cext ) n = N áCext ñ,
(3.10)
n =1
where áCext ñ is the average extinction cross section per particle. Analogously, by
substituting Eq. (3.8) in Eqs. (2.106)–(2.121) and assuming that particle positions are
sufficiently random that
[S n (rˆ , nˆ inc )]ij [S n′ (rˆ , nˆ inc )]∗kl exp[i(∆ n − ∆ n′ )]
Re
n′ ≠ n
n
n Re
[S n (rˆ , nˆ inc )]ij [S n (rˆ , nˆ inc )]∗kl ,
i, j , k , l = 1, 2
(3.11a)
n
and, if i ≠ k or j ≠ l ,
[S n (rˆ , nˆ inc )]ij [S n′ (rˆ , nˆ inc )]∗kl exp[i(∆ n − ∆ n′ )]
Im
n′ ≠ n
n
n Im
[S n (rˆ , nˆ inc )]ij [S n (rˆ , nˆ inc )]∗kl ,
(3.11b)
n
it is straightforward to show that the total phase matrix of the volume element is given
by
N
Z n = N á Zñ ,
Z=
(3.12)
n =1
where áZñ is the average phase matrix per particle. Equations (2.160) and (2.161)
then give the total scattering and absorption cross sections of the volume element:
3 Scattering by collections of independent particles
71
N
Csca =
(Csca ) n = N áCsca ñ ,
(3.13)
(Cabs ) n = N áCabs ñ,
(3.14)
n =1
N
Cabs =
n =1
where áCsca ñ is the average scattering cross section and áCabs ñ the average absorption cross section per particle. Finally, Eqs. (2.166), (2.169), (2.178), and (2.186)
yield
N
C pr =
(C pr ) n = N áC pr ñ ,
(3.15)
(Κ e ) n = N áΚ e ñ,
(3.16)
n =1
N
Κe =
n =1
where áCpr ñ is the average radiation-pressure cross section and áΚ e ñ the average
emission vector per particle. Thus, the optical cross sections, the phase and extinction
matrices, and the emission vector of the small volume element comprising randomly
positioned, widely separated particles are obtained by adding the respective optical
characteristics of the individual particles. Obviously, this property of additivity also
holds when the incident light is a parallel quasi-monochromatic beam rather than a
plane electromagnetic wave.
It should be recognized that at any given moment, the particles filling the volume
element form a certain spatial configuration, and the individual waves scattered by
different particles have a specific phase relation and interfere. However, even a minute displacement of the particles or a slight change in the scattering geometry during
the measurement may change the phase differences entirely. Therefore, in almost all
practical situations the light singly scattered by a collection of randomly positioned
particles and measured by a real detector appears to be incoherent, and the optical
characteristics of individual particles must be added without regard to phase.
Because the total phase matrix of the volume element is the sum of the phase matrices of the constituent particles, the nine independent quadratic relations between the
elements of the single-particle phase matrix as well as some quadratic inequalities
(see Section 2.6) generally no longer hold. Still, there are a number of linear and
quadratic inequalities that can be used for testing the elements of theoretically or experimentally obtained phase matrices of particle collections (Hovenier and van der
Mee 2000). The simplest and most important of them are
Z11 ≥ 0
(3.17)
(cf. Eqs. (2.106) and (3.12)) and
|Z ij | ≤ Z11,
i, j = 1, ..., 4.
(3.18)
Obviously, the reciprocity and symmetry relations (2.124) and (2.147a,b) remain
valid for the phase and extinction matrices of the volume element.
72
Scattering, Absorption, and Emission of Light by Small Particles
3.2
Ensemble averaging
Scattering media encountered in practice are usually mixtures of particles with different sizes, shapes, orientations, and refractive indices. Equations (3.9)–(3.16) imply
that theoretical computations of single scattering of light by a small volume element
consisting of such particles must include averaging the optical cross sections, the
phase and extinction matrices, and the emission vector over a representative particle
ensemble. The computation of ensemble averages is, in principle, rather straightforward. For example, for homogeneous ellipsoids with semi-axes a ∈ [a min , a max ],
b ∈ [bmin , bmax ], and c ∈ [cmin , cmax ] and the same refractive index, the ensembleaveraged phase matrix per particle is
a max
á Z(nˆ , nˆ ′ )ñ =
bmax
da
a min
2π
cmax
db
dc
bmin
cmin
π
dα
0
dβ sinβ
0
2π
dγ p (α , β , γ ; a, b, c)
0
× Z(nˆ , nˆ ′; α , β , γ ; a, b, c),
(3.19)
where the Euler angles α , β , and γ specify particle orientations with respect to the
laboratory reference frame, and p(α , β , γ ; a, b, c) is a probability density function
satisfying the normalization condition:
a max
bmax
da
a min
2π
c max
db
bmin
dc
cmin
dα
0
π
0
dβ sinβ
2π
dγ p (α , β , γ ; a, b, c) = 1.
(3.20)
0
The integrals in Eq. (3.19) are usually evaluated numerically by using appropriate
quadrature formulas. Some theoretical techniques (e.g., the T-matrix method) allow
analytical averaging over particle orientations, thereby bypassing time-consuming
integration over the Euler angles.
It is often assumed that the shape and size distribution and the orientation distribution are statistically independent. The total probability density function can then be
simplified by representing it as a product of two functions, one of which, ps (a, b, c),
describes the particle shape and size distribution and the other of which, po (α , β , γ ),
describes the distribution of particle orientations:
p (α , β , γ ; a, b, c) = ps (a, b, c) po (α , β , γ ).
(3.21)
Each is normalized to unity:
a max
bmax
da
a min
2π
0
dα
cmax
dc ps (a, b, c) = 1,
db
bmin
π
0
(3.22)
cmin
dβ sinβ
2π
dγ po (α , β , γ ) = 1.
(3.23)
0
In consequence, the problems of computing the shape and size and the orientation
averages are separated. Similarly, it is often convenient to separate averaging over
3 Scattering by collections of independent particles
73
shapes and sizes by assuming that particle shapes and sizes are statistically independent. For example, the shape of a spheroidal particle can be specified by its aspect
ratio ε (the ratio of the largest to the smallest axes) along with the designation of
either prolate or oblate, whereas the size of the particle can be specified by an
equivalent-sphere radius a. Then the shape and size probability density function
ps (ε , a ) can be represented as a product
ps (ε , a ) = p (ε )n(a ),
(3.24)
where p (ε ) describes the distribution of spheroid aspect ratios and n(a ) is the distribution of equivalent-sphere radii. Again, both p (ε ) and n(a ) are normalized to
unity:
ε max
dε p (ε ) = 1,
(3.25)
da n(a ) = 1.
(3.26)
ε min
a max
a min
In the absence of external forces such as magnetic, electrostatic, or aerodynamical
forces, all orientations of a nonspherical particle are equiprobable. In this practically
important case of randomly oriented particles, the orientation distribution function is
uniform with respect to the Euler angles of rotation, and we have
po, random (α , β , γ ) ≡
1
.
8π 2
(3.27)
An external force can make the orientation distribution axially symmetric, the axis of
symmetry being given by the direction of the force. For example, interstellar dust
grains can be axially oriented by a cosmic magnetic field (Martin 1978), whereas
nonspherical hydrometeors can be axially oriented by the aerodynamical force resulting from their non-zero falling velocity (Liou 1992). In this case it is convenient
to choose the laboratory reference frame with the z-axis along the external force direction so that the orientation distribution is uniform with respect to the Euler angles
α and γ :
po,axial (α , β , γ ) ≡
1
po ( β ).
4π 2
(3.28)
Particular details of the particle shape can also simplify the orientation distribution
function. For example, for rotationally symmetric bodies it is convenient to direct the
z-axis of the particle reference frame along the axis of rotation, in which case the orientation distribution function in the laboratory reference frame becomes independent
of the Euler angle γ :
po (α , β , γ ) ≡
1
po (α , β ).
2π
(3.29)
74
Scattering, Absorption, and Emission of Light by Small Particles
3.3
Condition of independent scattering
The inequalities (3.11a) and (3.11b) require the assumption that scattering is incoherent and that the positions of the particles filling the volume element are uncorrelated
during the time necessary to make the measurement. However, it is rather difficult to
give general and definitive criteria under which the inequalities (3.11a) and (3.11b)
are satisfied. Also, there is no obvious prescription for specifying the minimal interparticle separation that allows the use of the concept of the single-particle amplitude
scattering matrix in Eq. (3.3) and makes particles effectively independent scatterers.
Exact computations for a few specific cases can perhaps provide qualitative guidance.
Figure 3.1 shows the results of exact T-matrix computations (Chapter 5) of the ratios
Z 22 (ϑ sca , ϕ sca = 0; ϑ inc = 0, ϕ inc = 0)
,
Z11(ϑ sca , ϕ sca = 0; ϑ inc = 0, ϕ inc = 0)
−
Z 21(ϑ sca , ϕ sca = 0; ϑ inc = 0, ϕ inc = 0)
Z11(ϑ sca , ϕ sca = 0; ϑ inc = 0, ϕ inc = 0)
(3.30)
versus ϑ sca for randomly oriented two-sphere clusters with touching or separated
components (Mishchenko et al. 1995). The relative refractive index is 1.5 + i0.005,
the size parameter of the component spheres is k1a = 5, where k1 is the wave number
in the surrounding medium and a is the sphere radius, and the distance d between the
centers of the cluster components varies from 2a for touching spheres to 8a. For
comparison, in the lower panel the thin solid curve depicts the results for single scattering by independent spheres with k1a = 5 (regarding the upper panel, note that in
this case Z 22 Z11 ≡ 1; see Section 4.8). Obviously, the results for d = 8a are hardly
distinguishable from those for the independently scattering spheres. Even as small a
distance between the component sphere centers as four times their radii combined
with averaging over cluster orientations is already sufficient to reduce greatly the
near-field and interference effects and produce scattering patterns very similar to
those for the independent particles. For still larger spheres, with k1a = 15, the comparisons that can be made from Fig. 3.2 suggest qualitative independence at even
smaller separations. While these results with separation measured in terms of particle
size may be expected to become inapplicable for particles significantly smaller than a
wavelength, they suggest a simple approximate condition of independent scattering
by particles comparable to and larger than a wavelength.
3.4
Radiative transfer equation and coherent
backscattering
Let us now relax the requirement that the scattering medium be macroscopically small
and optically thin and be viewed from a distance much larger than its size. We thus
assume that N is so large that the condition N áCsca ñl −2 n 1 is violated, and the
3 Scattering by collections of independent particles
75
100
Z22 /Z11 (%)
75
50
d = 2a
d = 4a
d = 8a
Independent spheres
25
0
100
−Z21 /Z11 (%)
50
0
−50
−100
0
60
120
Scattering angle (deg)
180
Figure 3.1. The ratios Z 22 Z11 and − Z 21 Z11 (%) for randomly oriented two-sphere clusters
with touching (d = 2a ) and separated components and independently scattering spheres. The
component sphere size parameter k1a is 5 and the relative refractive index is 1.5 + i0.005.
contribution of multiply scattered light to the total signal scattered by the medium can
no longer be ignored. Furthermore, although the observation point is assumed to be
in the far-field zone of each constituent particle, it is not necessarily in the far-field
zone of the scattering medium as a whole, so that the observer may see scattered light
coming from different directions. A traditional approach in such cases is to assume
that scattering by different particles is still independent (which implies that the particles are randomly positioned and widely separated) and compute the characteristics of
76
Scattering, Absorption, and Emission of Light by Small Particles
100
Z22 /Z11 (%)
75
50
d = 2a
d = 4a
Independent spheres
25
0
100
−Z21 /Z11 (%)
50
0
−50
−100
0
60
120
Scattering angle (deg)
180
Figure 3.2. As in Fig. 3.1, but for k1a = 15.
multiply scattered radiation by solving the so-called radiative transfer equation
(RTE).
Radiative transfer theory originated as a phenomenological approach based on
considering the transport of energy through a medium filled with a large number of
particles and ensuring energy conservation (e.g., Chandrasekhar 1960; Sobolev 1974;
van de Hulst 1980; Apresyan and Kravtsov 1996; Lagendijk and van Tiggelen 1996;
Ishimaru 1997). It has been demonstrated, however, that the RTE can in fact be derived from the electromagnetic theory of multiple wave scattering in discrete random
media under certain simplifying assumptions (Mishchenko, 2002, 2003). This deriva-
3 Scattering by collections of independent particles
77
tion has clarified the physical meaning of the quantities entering the RTE and their
relation to single-scattering solutions of the Maxwell equations.
Instead of going into the details of this derivation, we will simply summarize the
main concepts of the phenomenological radiative transfer theory, present the conventional form of the RTE, and explain the meaning of the quantities appearing in this
equation. The electromagnetic radiation field at each point r inside the scattering
medium is approximated by a collection of quasi-monochromatic beams with a continuous distribution of propagation directions n̂ and angular frequencies ω and is
characterized by the local four-component monochromatic specific intensity column
vector
é I (r, nˆ , ω ) ù
ê
ú
Q(r, nˆ , ω ) ú
ê
I(r, nˆ , ω ) =
.
êU (r, nˆ , ω )ú
ú
ê
ëêV (r, nˆ , ω ) ûú
(3.31)
It is assumed that the elementary beams are incoherent and make independent contributions to I(r, nˆ , ω ). The elements Q(r, nˆ , ω ), U (r, nˆ , ω ), and V (r, nˆ , ω ) describe
the polarization state of light propagating in the direction n̂ at the observation point
specified by the position vector r, and the monochromatic specific intensity (or radiance) I (r, nˆ , ω ) is defined such that
I (r, nˆ , ω ) dω dt dS dΩ
(3.32)
is the amount of electromagnetic energy in an angular frequency interval (ω , ω + dω )
which is transported in a time interval dt through a surface element dS normal to n̂
and centered at r in directions confined to a solid angle element dΩ centered at the
direction of propagation n̂. All elements of the specific intensity vector have the
dimension of monochromatic radiance: energy per unit frequency interval per unit
time per unit area per unit solid angle.
In the phenomenological radiative transfer theory, a medium filled with a large number of discrete, sparsely and randomly distributed particles is treated as continuous and
locally homogeneous. The concept of single scattering and absorption by an individual
particle is thus replaced with the concept of single scattering and absorption by a small
homogeneous volume element. Furthermore, it is assumed that the result of scattering is
not the transformation of a plane incident wave into a spherical scattered wave but,
rather, the transformation of the specific intensity vector of the incident light into the
specific intensity vector of the scattered light:
I(r, nˆ sca , ω ) = Z(r, nˆ sca , nˆ inc , ω ) I(r, nˆ inc , ω ),
where Z(r, nˆ sca , nˆ inc , ω ) is the phase matrix of the small volume element.
An informal way to justify this assumption is to note that the product Z(nˆ sca , nˆ inc ) Iinc
in Eq. (2.102) may be interpreted as the scattered polarized power per unit solid angle.
78
Scattering, Absorption, and Emission of Light by Small Particles
Specifically, the polarized energy flow across a surface element ∆S normal to n̂ sca at a
distance r from the particle is given by ∆S r −2 Z(nˆ sca , nˆ inc ) Iinc and is, at the same time,
equal to the polarized power scattered within the solid angle element ∆Ω = ∆S r −2 centered at nˆ sca .
Another assumption in the phenomenological radiative transfer theory is that the
scattering, absorption, and emission characteristics of the small volume element follow
from the Maxwell equations and are given by the incoherent sums of the respective characteristics of the constituent particles according to Eqs. (3.9), (3.10), (3.12)–(3.14), and
(3.16).
The change in the specific intensity vector along the direction of propagation n̂ in a
medium containing sparsely and randomly distributed, arbitrarily oriented particles is
described by the following classical RTE (Rozenberg 1977; Tsang et al. 1985; Mishchenko 2002):
d
I(r, nˆ , ω ) = −n0 (r )áK(r, nˆ , ω )ñI(r, nˆ , ω )
ds
dnˆ ′ á Z(r, nˆ , nˆ ′, ω )ñI(r, nˆ ′, ω )
+ n0 (r )
4π
+ n0 (r )áK e [r, nˆ , T (r ), ω ]ñ,
(3.33)
where the path-length element ds is measured along the unit vector n̂, n0 (r ) is the
local particle number density, and áK(r, nˆ , ω )ñ,
á Z(r, nˆ , nˆ ′, ω )ñ, and
áK e [r, nˆ , T (r ), ω ]ñ are the local ensemble-averaged extinction and phase matrices
and emission vector per particle, respectively. The first term on the right-hand side of
Eq. (3.33) describes the change in the specific intensity vector over the distance ds
caused by extinction and dichroism, the second term describes the contribution of
light illuminating a small volume element centered at r from all directions nˆ ′ and
scattered into the direction n̂, and the third term describes the contribution of the
emitted light.
The radiative transfer equation must be supplemented by boundary conditions appropriate for a given physical problem. In particular, the boundary conditions must
correspond to the macroscopic geometry of the scattering medium and specify the
direction, polarization state, and frequency distribution of the external incident light.
For example, in the case of light scattering by the atmosphere, a standard model is a
plane-parallel particulate medium illuminated from above by a parallel beam representing solar radiation and bounded from below by a reflecting surface. The solution
of the RTE yields the specific intensity vector of the outgoing radiation at each
boundary point and, thereby, the angular distribution and polarization state of light
multiply scattered (reflected and transmitted) by the medium. It also provides the
specific intensity vector of the internal radiation field. General solutions of Eq. (3.33)
have been discussed by, e.g., Mishchenko (1990a) and Haferman (2000).
Despite the approximate character of the standard radiative transfer theory, it pro-
3 Scattering by collections of independent particles
n̂ obs
n̂ ill
n̂ obs
79
n̂ ill
r1
rN
…
r2
rN −1
Figure 3.3. Schematic explanation of coherent backscattering.
vides a powerful and reasonably general prescription for the treatment of the interaction of light with particulate media and is accordingly applicable to a broad range of
practical situations. However, owing to some of the basic assumptions in the development of the classical RTE, there are circumstances for which it is not sufficient.
For example, since the classical RTE does not take full account of interference effects
it does not describe directly the so-called coherent backscattering of light (otherwise
known as weak photon localization) (Watson 1969). To explain the physical origin of
this phenomenon, let us consider a layer composed of discrete, randomly positioned
scattering particles and illuminated by a parallel beam of light incident in the direction n̂ ill (Fig. 3.3). The distant observer measures the intensity of light reflected by
the layer in the direction nˆ obs. The reflected signal is composed of the contributions
made by waves scattered along various paths inside the layer involving different
combinations of particles. Let us consider the two conjugate scattering paths shown
in Fig. 3.3 by solid and broken lines. These paths go through the same group of N
particles, denoted by their positions r1 , r2 , …, rN , but in opposite directions. The
waves scattered along the two conjugate paths interfere, the interference being constructive or destructive depending on the phase difference
∆ = k1 (rN − r1) ⋅ (nˆ ill + nˆ obs ),
(3.34)
where k1 is the wave number in the surrounding medium. If the observation direction is far from the exact backscattering direction given by −nˆ ill , then the waves
scattered along conjugate paths involving different groups of particles interfere in
different ways, and the average effect of the interference is zero owing to the randomness of particle positions. Consequently, the observer measures some average,
incoherent intensity that is well described by the classical radiative transfer theory.
However, at exactly the backscattering direction (nˆ obs = −nˆ ill ), the phase difference
between conjugate paths involving any group of particles is identically equal to zero,
Eq. (3.34), and the interference is always constructive, thereby resulting in a coherent
80
Scattering, Absorption, and Emission of Light by Small Particles
backscattering intensity peak superposed on the incoherent background (Tsang et al.
1985; Barabanenkov et al. 1991; Sheng 1995). The scattering paths involving only
one particle do not have conjugate counterparts and do not contribute to the coherent
intensity peak. Kuga and Ishimaru (1984) were the first to detect coherent backscattering in a controlled laboratory experiment, although it may have been unknowingly
observed by Lyot (1929) in the form of the so-called coherent polarization opposition
effect (Mishchenko et al. 2000e).
An exact computation of the coherent backscattering effect based on solving the
Maxwell equations is feasible only for few-component clusters and is complicated by
several factors. First, the scattering pattern for a monodisperse cluster in a fixed orientation is always heavily burdened by multiple maxima and minima resulting from
the interference of partial waves scattered by the cluster components and by the intricate resonance structure of the single-scattering contribution (Section 9.1). Second,
the scattering pattern can be further affected by near-field effects that result from the
close proximity of the component particles. Third, simple trigonometry shows that
the angular width of the coherent backscattering intensity peak is of the order
1 k1 á d ñ = λ 1 2π á d ñ , where á d ñ is the average distance between the cluster components and λ 1 is the wavelength in the surrounding medium. Therefore, the peak may
be too broad to be identified reliably unless the cluster components are widely separated. However, increasing the distance between the cluster components diminishes
the contribution of multiple scattering and, thus, the amplitude of the coherent backscattering peak, thereby making it difficult to detect.
To smooth out the effect of the first factor and make the backscattering peak detectable, one must compute a scattering pattern that is averaged over particle sizes,
cluster orientations, and distances between the components. Furthermore, the average
distance between the cluster components must be much larger than the size of the
components and the wavelength but yet small enough that the multiple-scattering
contribution to the total signal is still significant.
Ismagilov and Kravtsov (1993) studied analytically the simplest case, two widely
separated spheres with diameters much smaller than the wavelength, but found that
the amplitude of the coherent backscattering intensity peak was extremely small because of the weakness of the multiple-scattering contribution to the total scattered
signal. Mishchenko (1996a) used the exact superposition T-matrix method (Section
5.9) to compute far-field scattering by polydisperse, randomly oriented clusters composed of two equal wavelength-sized spheres with varying center-to-center distances.
He computed the ratio of the intensity scattered by the clusters to the intensity scattered by two independent polydisperse spheres of the same average size, assuming
unpolarized incident light. Figure 3.4 shows this ratio versus scattering angle (the
angle between the incidence and scattering directions) calculated for k1 á d ñ = 25, average component sphere size parameter k1 á añ = 5, and relative refractive index m =
1.2. The curve clearly exhibits a backscattering enhancement with an angular width
comparable to 1 k1 á d ñ and an amplitude of about 1.03. Mishchenko (1996a) found
3 Scattering by collections of independent particles
81
Normalized intensity
1.05
1
0.95
160
170
Scattering angle (deg)
180
Figure 3.4. Coherent backscattering by polydisperse, randomly oriented two-sphere clusters.
that this feature persisted when k1 á d ñ, k1 á añ , and m were varied, thereby indicating
that it was indeed caused by coherent backscattering.
The amplitude of the coherent backscattering peak (the ratio of the intensity in the
center of the peak to the background value) can be significantly greater for very large
collections of particles because of the much stronger contribution of multiple scattering (van Albada and Lagendijk 1985; Wolf et al. 1988; Labeyrie et al. 2000). For
example, the measurement results depicted in Fig. 3.5 show an amplitude of almost
1.8. Unfortunately, the exact theory of coherent backscattering for large particle collections is extremely complicated and has been developed only for the case of reflection of light by a semi-infinite layer composed of nonabsorbing particles with sizes
much smaller than the wavelength (Ozrin 1992; Amic et al. 1997). An exact result
was obtained by Mishchenko (1992b), who used the reciprocity relation of Eq.
(2.124) to show that the photometric and polarization characteristics of coherent
backscattering at exactly the backscattering direction as well as outside the backscattering peak can be expressed in terms of the solution of the classical RTE. Other
theoretical approaches are based on the so-called diffusion approximation (Stephen
and Cwilich 1986) and the Monte Carlo technique (van Albada and Lagendijk 1987;
Martinez and Maynard 1994; Iwai et al. 1995).
Because the angular width of the intensity peak caused by coherent backscattering
from optically thick layers is proportional to the ratio of the wavelength to the photon
mean free path, it is negligibly small for sparse particle collections and does not affect
the results of remote sensing observations of such tenuous objects as clouds, aerosols,
and precipitation. However, measurements of coherent backscattering have proved to
82
Scattering, Absorption, and Emission of Light by Small Particles
2
Enhacement factor
1.8
1.6
1.4
1.2
1
20
10
0
10
Phase angle (mrad)
20
Figure 3.5. Angular profile of the coherent backscattering peak produced by a 1500-µm-thick
slab of 9.6 vol% of 0.215-µm-diameter polystyrene spheres suspended in water. The slab was
illuminated by a linearly polarized laser beam ( λ 1 = 633 nm ) incident normally to the slab
surface. The scattering plane (i.e., the plane through the vectors n̂ ill and nˆ obs, Fig. 3.3) was
fixed in such a way that the electric vector of the incident beam vibrated in this plane. The
detector measured the component of the backscattered intensity polarized parallel to the scattering plane. The curve shows the profile of the backscattered intensity normalized by the intensity of the incoherent background as a function of the phase angle. The latter is defined as
the angle between the vectors n̂ obs and −nˆ ill. (After van Albada et al. 1987.)
be a valuable characterization tool in laboratory and remote sensing studies of layers
composed of more closely spaced particles, such as particle suspensions and natural
and artificial particulate surfaces (e.g., Muinonen 1993; Shkuratov 1994; Mishchenko
1996b; POAN Research Group 1998; Lenke and Maret 2000).
Further reading
A detailed discussion of the concept of single scattering by a small volume element
was recently presented by Mishchenko et al. (2004a).
Chapter 4
Scattering matrix and macroscopically isotropic
and mirror-symmetric scattering media
By definition, the phase matrix relates the Stokes parameters of the incident and scattered beams, defined relative to their respective meridional planes. In contrast to the
phase matrix, the scattering matrix F relates the Stokes parameters of the incident and
scattered beams defined with respect to the scattering plane, that is, the plane through
the unit vectors n̂ inc and n̂ sca (van de Hulst 1957).
A simple way to introduce the scattering matrix is to direct the z-axis of the reference frame along the incident beam and superpose the meridional plane with ϕ = 0
and the scattering plane (Fig. 4.1). Then the scattering matrix F can be defined as
F(ϑ sca ) = Z(ϑ sca , ϕ sca = 0; ϑ inc = 0, ϕ inc = 0).
(4.1)
In general, all 16 elements of the scattering matrix are non-zero and depend on the
particle orientation with respect to the incident and scattered beams.
The choice of laboratory reference frame, with z-axis along the incidence direction
and the xz-plane with x ≥ 0 coinciding with the scattering plane, can often be inconvenient because any change in the incidence direction and/or orientation of the scattering plane also changes the orientation of the scattering particle with respect to the
coordinate system. However, we will show in this chapter that the concept of the
scattering matrix can be very useful in application to so-called macroscopically isotropic and mirror-symmetric scattering media, because the scattering matrix of such a
particle collection becomes independent of incidence direction and orientation of the
scattering plane, depends only on the angle Θ = arccos(nˆ inc ⋅ nˆ sca ) between the incidence and scattering directions, and has a simple block-diagonal structure.
83
84
Scattering, Absorption, and Emission of Light by Small Particles
z
n̂ inc
n̂ sca
⊕
Θ
⊕
O
x
Bisectrix
− n̂
sca
− n̂ inc
Figure 4.1. The xz-plane of the reference frame acts as the scattering plane. The arrows perpendicular to the unit n̂ - vectors show the corresponding unit ϑ̂ - vectors. The symbols ⊕ and
⊙ indicate the corresponding unit ϕ̂ - vectors, which are directed into and out of the paper,
respectively.
4.1
Symmetries of the Stokes scattering matrix
We begin by considering special symmetry properties of the amplitude scattering
matrix that exist when both the incidence and the scattering directions lie in the xzplane (van de Hulst 1957). For the particle shown schematically in Fig. 4.2(a), let
é S11
ê
ë S 21
S12 ù
ú
S 22 û
(4.2a)
be the amplitude scattering matrix that corresponds to the directions of incidence and
scattering given by n̂ inc and nˆ sca , respectively (Fig. 4.1). Rotating this particle by
180º about the bisectrix (i.e., the line in the scattering plane that bisects the angle
π − Θ between the unit vectors − n̂ inc and n̂ sca in Fig. 4.1) puts it in the orientation
schematically shown in Fig. 4.2(b). It is clear that the amplitude scattering matrix
(4.2a) is also the amplitude scattering matrix for this rotated particle when the directions of incidence and scattering are given by − n̂ sca and − nˆ inc, respectively. Therefore, the reciprocity relation (2.64) implies that the amplitude scattering matrix of the
4 Scattering matrix and isotropic and mirror-symmetric media
(a)
(b)
(c)
85
(d)
Figure 4.2. Two orientations of an arbitrary particle and two orientations of its mirrorsymmetric particle that give rise to certain symmetries in scattering patterns. (After van de
Hulst 1957.)
particle shown in Fig. 4.2(b) that corresponds to the original directions of incidence
and scattering, n̂ inc and nˆ sca , is simply
é S11
ê
ë− S12
− S 21 ù
ú.
S 22 û
(4.2b)
Mirroring the original particle, Fig. 4.2(a), with respect to the scattering plane
gives the particle shown in Fig. 4.2(c). If we also reversed the direction of the unit
vectors ϕ̂ inc and ϕ̂ sca in Fig. (4.1), then we would have the same scattering problem
as for the particle shown in Fig. 4.2(a). We may thus conclude that the amplitude
scattering matrix for the particle shown in Fig. 4.2(c) that corresponds to the directions of incidence and scattering n̂ inc and n̂ sca is
é S11
ê
ë− S 21
− S12 ù
ú.
S 22 û
(4.2c)
Finally, mirroring the original particle with respect to the bisectrix plane (i.e., the
plane through the bisectrix and the y-axis) gives the particle shown in Fig. 4.2(d).
Since this particle is simply the mirror-symmetric counterpart of the particle shown in
Fig. 4.2(b), its amplitude scattering matrix corresponding to the directions of incidence and scattering n̂ inc and n̂ sca is
é S11
ê
ë S12
S 21 ù
ú.
S 22 û
(4.2d)
It can be seen that any two of the three transformations shown in Figs. 4.2(b)–4.2(d)
give the third.
We will now discuss the implications of Eqs. (4.2a)–(4.2d) for Stokes scattering
matrices of collections of independently scattering particles, by considering the following four examples (van de Hulst 1957).
(1) Let us first assume that a small volume element contains only one kind of particle and that each particle in a specific position, say Fig. 4.2(a), is accompanied by a
particle in the reciprocal position, Fig. 4.2(b). It then follows from Eqs. (2.106)–
86
Scattering, Absorption, and Emission of Light by Small Particles
(2.121), (3.12), (4.1), (4.2a), and (4.2b) that the scattering matrix of the small volume
element has the following symmetry:
é F11
ê
ê F12
ê− F13
ê
êë F14
F12
F13
F22
F23
− F23
F33
F24
− F34
F14 ù
ú
F24 ú
.
F34 ú
ú
F44 úû
(4.3)
The number of independent matrix elements is 10.
(2) As a second example, let us assume that the volume element contains particles
and their mirror-symmetric counterparts such that for each particle in orientation (a) a
mirror-symmetric particle in orientation (c) is present (Fig. 4.2). This excludes, for
example, scattering media composed of only right-handed or only left-handed helices.
It is easy to verify that the resulting scattering matrix involves eight independent elements and has the following structure:
é F11
ê
ê F21
ê 0
ê
êë 0
F12
0
F22
0
0
F33
0
F43
0 ù
ú
0 ú
.
F34 ú
ú
F44 úû
(4.4)
(3) As a third example, consider a volume element containing particles and their
mirror-symmetric counterparts and assume that any particle in orientation (a) is accompanied by a mirror-symmetric particle in orientation (d), Fig. 4.2. The scattering
matrix becomes
é F11
ê
ê F12
ê F13
ê
êë− F14
F12
F13
F22
F23
F23
F33
− F24
− F34
F14 ù
ú
F24 ú
F34 ú
ú
F44 úû
(4.5)
and has 10 independent elements.
(4) Finally, consider a volume element containing particles and their mirrorsymmetric counterparts and make any two of the preceding assumptions. The third
assumption follows automatically, so that there are equal numbers of particles in orientations (a), (b), (c), and (d). The resulting scattering matrix is
é F11
ê
ê F12
ê0
ê
ëê 0
F12
0
F22
0
0
F33
0
− F34
0 ù
ú
0 ú
F34 ú
ú
F44 ûú
and has eight non-zero elements, of which only six are independent.
(4.6)
4 Scattering matrix and isotropic and mirror-symmetric media
4.2
87
Macroscopically isotropic and mirror-symmetric
scattering medium
Now we are ready to consider scattering by a small volume element containing randomly oriented particles. This means that there are many particles of each type and
their orientation distribution is uniform (see Eq. (3.27)). In this case the assumptions
of example (1) from the previous section are satisfied, and the total scattering matrix
is given by Eq. (4.3). Furthermore, if particles and their mirror-symmetric counterparts are present in equal numbers or each particle has a plane of symmetry, then the
assumptions of example 4 are satisfied, and the resulting scattering matrix is given by
Eq. (4.6).
As a consequence of random particle orientation, the scattering medium is macroscopically isotropic (i.e., there is no preferred propagation direction and no preferred
plane through the incidence direction). Therefore, the scattering matrix becomes independent of the incidence direction and the orientation of the scattering plane and
depends only on the angle between the incidence and scattering directions, that is, the
scattering angle
Θ = arccos(nˆ inc ⋅ nˆ sca ),
Θ ∈ [0, π ].
Furthermore, the assumptions of example (4) ensure that the scattering medium is
macroscopically mirror-symmetric with respect to any plane and make the structure
of the scattering matrix especially simple. Therefore, scattering media composed of
equal numbers of randomly oriented particles and their mirror-symmetric counterparts
and/or randomly oriented particles having a plane of symmetry can be called macroscopically isotropic and mirror-symmetric. Although this type of scattering medium
might be thought to be a rather special case, it nonetheless provides a very good numerical description of the scattering properties of many particle collections encountered in practice and is by far the most often used theoretical model. To emphasize
that the scattering matrix of a macroscopically isotropic and mirror-symmetric scattering medium depends only on the scattering angle, we rewrite Eq. (4.6) as
0
0 ù
é F11(Θ ) F12 (Θ )
ê
ú
0
0 ú
F12 (Θ ) F22 (Θ )
ê
F(Θ ) =
= N áF(Θ )ñ,
ê 0
0
F33(Θ ) F34 (Θ ) ú
ê
ú
− F34 (Θ ) F44 (Θ )úû
0
êë 0
(4.7)
where N is the number of particles in the volume element and áF(Θ )ñ is the ensemble-averaged scattering matrix per particle.
As a direct consequence of Eqs. (3.17) and (3.18) we have the inequalities
F11 ≥ 0,
| Fij | ≤ F11,
(4.8)
i, j = 1, K , 4.
(4.9)
Scattering, Absorption, and Emission of Light by Small Particles
88
Additional general inequalities for the elements of the scattering matrix (4.7) are
( F33 + F44 ) 2 + 4 F342 ≤ ( F11 + F22 ) 2 − 4 F122 ,
(4.10)
| F33 − F44 | ≤ F11 − F22 ,
(4.11)
| F22 − F12 | ≤ F11 − F12 ,
(4.12)
| F22 + F12 | ≤ F11 + F12.
(4.13)
The proof of these and other useful inequalities is given in Hovenier et al. (1986).
4.3
Phase matrix
Knowledge of the matrix F(Θ ) can be used to calculate the Stokes phase matrix for a
macroscopically isotropic and mirror-symmetric scattering medium. Assume that
0 < ϕ sca − ϕ inc < π and consider phase matrices Z(ϑ sca , ϕ sca ; ϑ inc , ϕ inc ) and Z(ϑ sca ,
ϕ inc; ϑ inc , ϕ sca ). The second matrix involves the same zenith angles of the incident
and scattered beams as the first, but the azimuth angles are switched, as indicated in
their respective scattering geometries; these are shown in Figs. 4.3(a), (b). The phase
matrix links the Stokes vectors of the incident and scattered beams, specified relative
to their respective meridional planes. Therefore, to compute the Stokes vector of the
scattered beam with respect to its meridional plane, we must
●
●
●
calculate the Stokes vector of the incident beam with respect to the scattering
plane;
multiply it by the scattering matrix, thereby obtaining the Stokes vector of the
scattered beam with respect to the scattering plane; and finally
compute the Stokes vector of the scattered beam with respect to its meridional
plane (Chandrasekhar 1960).
This procedure involves two rotations of the reference plane, as shown in Figs. 4.3(a),
(b), and yields
Z(ϑ sca , ϕ sca ; ϑ inc , ϕ inc ) = L(−σ 2 )F(Θ )L(π − σ 1)
C1 F12 (Θ )
S1 F12 (Θ )
0
ù
é F11(Θ )
ú
ê
C 2 F12 (Θ )
C1C 2 F22 (Θ ) − S1 S 2 F33(Θ )
S1C 2 F22 (Θ ) + C1 S 2 F33 (Θ ) S 2 F34 (Θ ) ú
ê
,
=
ê− S 2 F12 (Θ ) − C1 S 2 F22 (Θ ) − S1C 2 F33 (Θ ) − S1 S 2 F22 (Θ ) + C1C 2 F33(Θ ) C 2 F34 (Θ )ú
ú
ê
S1 F34 (Θ )
F44 (Θ ) ûú
0
− C1 F34 (Θ )
ëê
(4.14)
4 Scattering matrix and isotropic and mirror-symmetric media
z
n̂ sca
σ2
σ1
n̂ inc
Θ
y
(a)
x
z
n̂ sca
σ2
σ1
Θ
n̂inc
y
(b)
x
Figure 4.3. Illustration of the relationship between the phase and scattering matrices.
89
Scattering, Absorption, and Emission of Light by Small Particles
90
Z(ϑ sca , ϕ inc; ϑ inc , ϕ sca ) = L(σ 2 − π )F(Θ )L(σ 1)
0
C1 F12 (Θ )
− S1 F12 (Θ )
é F11(Θ )
ù
ê
ú
C 2 F12 (Θ ) C1C 2 F22 (Θ ) − S1 S 2 F33 (Θ ) − S1C 2 F22 (Θ ) − C1 S 2 F33 (Θ ) − S 2 F34 (Θ )ú
ê
,
=
ê S 2 F12 (Θ ) C1 S 2 F22 (Θ ) + S1C 2 F33 (Θ ) − S1 S 2 F22 (Θ ) + C1C 2 F33 (Θ ) C 2 F34 (Θ ) ú
ê
ú
0
F44 (Θ ) ûú
− S1 F34 (Θ )
− C1 F34 (Θ )
ëê
(4.15)
where
Ci = cos 2σ i ,
S i = sin 2σ i ,
i = 1, 2,
(4.16)
and the rotation matrix L is defined by Eq. (1.97). (Recall that a rotation angle is
positive if the rotation is performed in the clockwise direction when one is looking in
the direction of propagation; see Section 1.5.) The scattering angle Θ and the angles
σ 1 and σ 2 can be calculated from ϑ sca , ϑ inc, ϕ sca , and ϕ inc using spherical trigonometry:
cosΘ = cosϑ sca cosϑ inc + sinϑ sca sinϑ inc cos(ϕ sca − ϕ inc ),
(4.17)
cos σ 1 =
cosϑ sca − cosϑ inc cosΘ
,
sinϑ inc sinΘ
(4.18)
cos σ 2 =
cosϑ inc − cosϑ sca cosΘ
.
sinϑ sca sinΘ
(4.19)
Equations (4.14)–(4.19) demonstrate the obvious fact that the phase matrix of a macroscopically isotropic and mirror-symmetric medium depends only on the difference
between the azimuthal angles of the scattered and incident beams rather than on their
specific values. Comparison of Eqs. (4.14) and (4.15) yields the symmetry relation
(Hovenier 1969):
Z(ϑ sca , ϕ inc; ϑ inc , ϕ sca ) = ∆ 34 Z(ϑ sca , ϕ sca ; ϑ inc , ϕ inc ) ∆ 34 ,
(4.20)
where
T
−1
= ∆ 34
∆ 34 = ∆ 34
é1
ê
0
=ê
ê0
ê
êë0
0ù
ú
1 0
0ú
.
0 −1 0 ú
ú
0 0 − 1úû
0
0
(4.21)
It is also easy to see from either Eq. (4.14) or Eq. (4.15) that (Hovenier 1969)
Z(π − ϑ sca , ϕ sca ; π − ϑ inc , ϕ inc ) = ∆ 34 Z(ϑ sca , ϕ sca ; ϑ inc , ϕ inc ) ∆ 34 ,
(4.22)
which is a manifestation of symmetry with respect to the xy-plane. Finally, we can
verify that
Z(π − ϑ inc , ϕ inc + π ; π − ϑ sca , ϕ sca + π ) = ∆ 3 [Z (ϑ sca , ϕ sca ; ϑ inc , ϕ inc )]T ∆ 3,
(4.23)
4 Scattering matrix and isotropic and mirror-symmetric media
91
where the matrix ∆ 3 is given by Eq. (2.125). Obviously, this is the reciprocity relation (2.124). Other symmetry relations can be derived by forming combinations of
Eqs. (4.20), (4.22), and (4.23). For example, combining Eqs. (4.20) and (4.22) yields
Z(π − ϑ sca , ϕ inc; π − ϑ inc, ϕ sca ) = Z(ϑ sca , ϕ sca ; ϑ inc, ϕ inc ).
(4.24)
Although Eq. (4.14) is valid only for 0 < ϕ sca − ϕ inc < π , combining it with Eq.
(4.20) yields the phase matrix for all possible incidence and scattering directions. The
symmetry relations (4.22) and (4.23) further reduce the range of independent scattering geometries and can be very helpful in theoretical calculations or consistency
checks on measurements.
4.4
Forward-scattering direction and extinction matrix
By virtue of spatial isotropy, the extinction matrix of a macroscopically isotropic and
mirror-symmetric medium is independent of the direction of light propagation and
orientation of the reference plane used to define the Stokes parameters. It also follows from Eqs. (2.142)–(2.145), (3.9), and (4.2a)–(4.2d) that Κ 13 = Κ 14 = Κ 23 = Κ 24 =
Κ 31 = Κ 32 = Κ 41 = Κ 42 = 0. Furthermore, we are about to show that the remaining offdiagonal elements of the extinction matrix also vanish.
We will assume for simplicity that light is incident along the positive direction of
the z-axis of the laboratory reference frame and will use the xz-plane with x ≥ 0 as
the meridional plane of the incident beam. We will also assume that the initial orientation of a particle is such that the particle reference frame coincides with the laboratory reference frame. The forward-scattering amplitude matrix of the particle in the
initial orientation computed in the laboratory reference frame is thus equal to the forward-scattering amplitude matrix computed in the particle reference frame. We will
denote the latter as S P. Let us now rotate the particle through an Euler angle α
about the z-axis in the clockwise direction as viewed in the positive z-direction (Figs.
2.2 and 4.4) and denote the forward-scattering amplitude matrix of this rotated particle with respect to the laboratory reference frame as SαL . This matrix relates the column of the electric field vector components of the incident field to that of the field
scattered in the exact forward direction:
é EϑscaL ù
ê sca ú ∝ SαL
êë Eϕ L úû
é EϑincL ù
ê inc ú,
êë Eϕ L úû
(4.25)
where the subscript L indicates that all field components are computed in the laboratory reference frame. Figure 4.4 shows the directions of the respective unit ϑ̂ - and
ϕ̂ - vectors for the incident and the forward-scattered beams. Simple trigonometry
allows us to express the column of the electric vector components in the particle reference frame in terms of that in the laboratory reference frame by means of a trivial
Scattering, Absorption, and Emission of Light by Small Particles
92
y
y′
ϕˆ Pinc,
ϕˆ Psca
x′
ϕˆ Linc, ϕˆ Lsca
α
ϑˆ Pinc, ϑˆ Psca
α
x
ϑˆ Linc, ϑˆ Lsca
Figure 4.4. Rotation of the particle through an Euler angle α about the z-axis transforms the
laboratory reference frame L{x, y, z} into the particle reference frame P{x′, y ′, z}. Since both
the incident and the scattered beams propagate in the positive z-direction, their respective unit
ϑ̂ - and ϕ̂ - vectors are the same.
matrix multiplication (cf. Fig. 4.4):
é EϑincP ù é C S ù é EϑincL ù
ê inc ú = ê
ú ê inc ú,
ëê Eϕ P ûú ë− S C û ëê Eϕ L ûú
(4.26)
where C = cosα and S = sin α . Conversely,
é Eϑsca
éC
L ù
ê sca ú = ê
êë Eϕ L úû ë S
− S ù é Eϑsca
P ù
ú ê sca ú.
C û êë Eϕ P úû
(4.27)
Rewriting Eq. (4.25) in the particle reference frame,
é Eϑsca
é EϑincP ù
P ù
S
∝
ê sca ú
P ê inc ú ,
êë Eϕ P úû
êë Eϕ P úû
(4.28)
and using Eqs. (4.26) and (4.27), we finally derive
éC
SαL = ê
ëS
− Sù é C
úS P ê
C û ë− S
Sù
ú
Cû
éC 2 S11P − SCS12 P − SCS 21P + S 2 S 22 P
=ê
êë SCS11P − S 2 S12 P + C 2 S 21P − SCS 22 P
SCS11P + C 2 S12 P − S 2 S 21P − SCS 22 P ù
ú.
S 2 S11P + SCS12 P + SCS 21P + C 2 S 22 P úû
(4.29)
For α = 0 and α = π 2 ,
é S11P S12 P ù
S 0L = ê
ú,
ë S 21P S 22 P û
é S 22 P − S 21P ù
SπL 2 = ê
ú.
ë− S12 P S11P û
(4.30)
(4.31)
4 Scattering matrix and isotropic and mirror-symmetric media
93
Because we are assuming random orientation of the particles in the small volume
element, for each particle in the initial orientation, α = 0, there is always a particle of
the same type but in the orientation corresponding to α = π 2 . It, therefore, follows
from Eqs. (2.141), (2.146), (3.9), (4.30), and (4.31) that Κ 12 = Κ 21 = Κ 34 = Κ 43 = 0.
Finally, recalling Eq. (2.159), we conclude that the extinction matrix of a small volume element containing equal numbers of randomly oriented particles and their mirror-symmetric counterparts and/or randomly oriented particles having a plane of
symmetry is diagonal:
Κ(nˆ ) ≡ Κ = Cext ∆ = N áCext ñ ∆ ,
(4.32)
where
é1
ê
0
∆=ê
ê0
ê
êë0
0 0 0ù
ú
1 0 0ú
0 1 0ú
ú
0 0 1úû
is the 4× 4 unit matrix, N is the number of particles in the volume element, and
áCext ñ is the average extinction cross section per particle, which is now independent
of the direction of propagation and polarization state of the incident light. This significant simplification is useful in many practical circumstances.
The scattering matrix also becomes simpler when Θ = 0. From Eqs. (2.107),
(2.110), (2.117), (2.120), (4.30), and (4.31), we find that F12 (0) = F21(0) = F34 (0) =
F43(0) = 0. Equation (4.29) gives for α = π 4:
SπL 4 =
1 é S11P − S12 P − S 21P + S 22 P
ê
2 ë S11P − S12 P + S 21P − S 22 P
S11P + S12 P − S 21P − S 22 P ù
ú.
S11P + S12 P + S 21P + S 22 P û
(4.33)
Equations (2.111), (2.116), (4.30), and (4.33) and a considerable amount of algebra
yield F22 (0) = F33(0). Thus, recalling Eq. (4.7), we find that the forward-scattering
matrix for a macroscopically isotropic and mirror-symmetric medium is diagonal and
has only three independent elements:
0
0
0 ù
é F11(0)
ê
ú
0
0
0 ú
F22 (0)
ê
F(0) =
ê 0
0
0 ú
F22 (0)
ê
ú
0
0
F44 (0)úû
êë 0
(4.34)
(van de Hulst 1957).
Rotationally-symmetric particles are obviously mirror-symmetric with respect to
the plane through the direction of propagation and the axis of symmetry. Choosing
this plane as the x′z ′ - plane of the particle reference frame, we see from Eq. (4.2c)
that S12 P = S 21P = 0. This simplifies the amplitude scattering matrices (4.30) and
(4.33) and ultimately yields
Scattering, Absorption, and Emission of Light by Small Particles
94
F44 (0) = 2 F22 (0) − F11(0),
0 ≤ F22 (0) ≤ F11(0)
(4.35)
(Mishchenko and Travis 1994c; Hovenier and Mackowski 1998).
4.5
Backward scattering
Equation (4.1) provides an unambiguous definition of the scattering matrix in terms
of the phase matrix, except for the exact backscattering direction. Indeed, the backscattering direction for an incidence direction (ϑ inc, ϕ inc ) is given by
(π − ϑ inc, ϕ inc + π ). Therefore, the complete definition of the scattering matrix should
be as follows:
F(ϑ
sca
ìïZ(ϑ sca , 0; 0, 0)
)=í
ïî Z(π , π ; 0, 0)
for ϑ sca ∈ [0, π ),
for ϑ sca = π ,
which seems to be different from Eq. (4.1). It is easy to see, however, that
Z(π , 0; 0, 0) = L(π )Z(π , π ; 0, 0) ≡ Z(π , π ; 0, 0), cf. Eq. (1.97), which demonstrates
the equivalence of the two definitions.
We are ready now to consider the case of scattering in the exact backward direction, using the complete definition of the scattering matrix and the backscattering
theorem derived in Section 2.3. Let us assume that light is incident along the positive
z-axis of the laboratory coordinate system and is scattered in the opposite direction;
we use the xz-plane with x ≥ 0 as the meridional plane of the incident beam. As in
the previous section, we consider two particle orientations relative to the laboratory
reference frame: (i) the initial orientation, when the particle reference frame coincides
with the laboratory reference frame, and (ii) the orientation obtained by rotating the
particle about the z-axis through a positive Euler angle α . Figure 4.5 shows the respective unit ϑ̂ - and ϕ̂ - vectors for the incident and the backscattered beams. Denote the backscattering amplitude matrix in the particle reference frame as S P and
the backscattering amplitude matrix in the laboratory reference frame for the rotated
particle as SαL . A derivation similar to that in the previous section gives
é C Sù
é C Sù
SαL = ê
ú
ú SP ê
ë− S C û
ë− S C û
é C 2 S11P − SCS12 P + SCS 21P − S 2 S 22 P
=ê
2
2
ëê− SCS11P + S S12 P + C S 21P − SCS 22 P
SCS11P + C 2 S12 P + S 2 S 21P + SCS 22 P ù
ú.
− S 2 S11P − SCS12 P + SCS 21P + C 2 S 22 P ûú
(4.36)
This formula can be simplified, because the backscattering theorem (2.65) yields
S 21P = − S12 P . Assuming that particles are randomly oriented and considering the
cases α = 0 and α = π 2 , we find that F12 (π ) = F21(π ) = F34 (π ) = F43(π ) = 0.
Similarly, considering the cases α = 0 and α = π 4 yields F33(π ) = − F22 (π ). Finally, recalling Eqs. (2.131) and (4.7), we conclude that the backscattering matrix for
4 Scattering matrix and isotropic and mirror-symmetric media
95
y
y′
x′
ϕˆ Linc
ϕˆ Pinc
α
ϑˆ Pinc, ϑˆ Psca
α
x
ϑˆ Linc, ϑˆ Lsca
α
ϕˆ Lsca
ϕˆ Psca
Figure 4.5. As in Fig. 4.4, but for the case of scattering in the exact backward direction.
a macroscopically isotropic and mirror-symmetric medium is diagonal and has only
two independent elements:
0
0
0
é F11(π )
ù
ê
ú
0
0
0
F22 (π )
ê
ú
F(π ) =
ê 0
ú
0
− F22 (π )
0
ê
ú
0
0
F11(π ) − 2 F22 (π )úû
êë 0
(4.37)
(Mishchenko and Hovenier 1995). According to Eq. (4.9) F44 ≤ F11, so we always
have
F22 (π ) ≥ 0.
4.6
(4.38)
Scattering cross section, asymmetry parameter, and
radiation pressure
Like all other macroscopic scattering characteristics, the average scattering cross section per particle for a macroscopically isotropic and mirror-symmetric medium is independent of the direction of illumination. Therefore, we will evaluate the integral on
the right-hand side of Eq. (2.160) assuming that the incident light propagates along
the positive z-axis of the laboratory reference frame and that the xz-plane with x ≥ 0
is the meridional plane of the incident beam. Figure 4.6 shows that in order to compute the Stokes vector of the scattered beam with respect to its own meridional plane,
we must rotate the reference frame of the incident beam by the angle ϕ , thereby
modifying the Stokes vector of the incident light according to Eq. (1.97) with η = ϕ ,
96
Scattering, Absorption, and Emission of Light by Small Particles
z
n̂ inc
n̂ sca
ϑ
y
ϕ
x
Figure 4.6. Illustration of the relationship between the phase and scattering matrices when the
incident light propagates along the positive z-axis.
and then multiply the new Stokes vector of the incident light by the scattering matrix.
Therefore, the average phase matrix per particle is simply
á Z(nˆ sca , nˆ inc )ñ = áF(ϑ )ñL(ϕ )
é á F11(ϑ )ñ á F12 (ϑ )ñ cos 2ϕ
ê
á F12 (ϑ )ñ á F22 (ϑ )ñ cos 2ϕ
=ê
ê 0
á F33(ϑ )ñ sin 2ϕ
ê
− á F34 (ϑ )ñ sin 2ϕ
êë 0
− á F12 (ϑ )ñ sin 2ϕ
− á F22 (ϑ )ñ sin 2ϕ
á F33(ϑ )ñ cos 2ϕ
− á F34 (ϑ )ñ cos 2ϕ
ù
ú
0 ú
.
á F34 (ϑ )ñ ú
ú
á F44 (ϑ )ñ úû
0
(4.39)
Substituting this formula in Eq. (2.160), we find that the average scattering cross section per particle is independent of the polarization state of the incident light and is
given by
áCsca ñ = 2π
π
dϑ sin ϑ á F11(ϑ )ñ.
(4.40)
0
The ensemble-averaged asymmetry parameter must also be independent of nˆ inc , and
Eqs. (2.166), (2.169), and (4.39) yield
ácosΘ ñ =
2π
áCsca ñ
π
dϑ sin ϑ cos ϑ á F11(ϑ )ñ.
(4.41)
0
Obviously, ácos Θ ñ is polarization-independent. Equations (2.176), (4.39), and
(4.41) show that the average radiation force per particle is now directed along nˆ inc:
4 Scattering matrix and isotropic and mirror-symmetric media
á Fñ =
97
1 inc inc
nˆ I [áCext ñ − áCsca ñá cos Θ ñ ]
c
1 inc inc
nˆ I áCpr ñ,
c
(4.42)
áCpr ñ = áCext ñ − áCsca ñá cos Θ ñ
(4.43)
=
where
is the polarization- and direction-independent average radiation-pressure cross section
per particle. The average absorption cross section per particle,
áCabs ñ = áCext ñ − áCsca ñ,
(4.44)
and the average single-scattering albedo,
ϖ=
áCsca ñ
,
áCext ñ
(4.45)
are also independent of the direction and polarization of the incident beam. The
same, of course, is true of the extinction, scattering, absorption, and radiation pressure
efficiency factors, defined as
Qext =
áCext ñ
,
áG ñ
Qsca =
áCsca ñ
,
áG ñ
Qabs =
áCabs ñ
,
áG ñ
Qpr =
áC pr ñ
áG ñ
,
(4.46)
respectively, where áGñ is the average projection area per particle.
4.7
Thermal emission
Because the ensemble-averaged emission vector for a macroscopically isotropic and
mirror-symmetric medium must be independent of the emission direction, we will
calculate the integral on the right-hand side of Eq. (2.186) for light emitted in the
positive direction of the z-axis and will use the meridional plane ϕ = 0 as the reference plane for defining the emission Stokes vector. It is then obvious from Fig. 4.7
that the corresponding average phase matrix per particle can be calculated as
á Z (nˆ , nˆ ′ )ñ = L(−ϕ ′ )áF(ϑ ′ )ñ
á F11(ϑ ′ )ñ
á F12 (ϑ ′ )ñ
0
0
é
ù
ê
ú
á F12 (ϑ ′ )ñ cos 2ϕ ′ á F22 (ϑ ′ )ñ cos 2ϕ ′ á F33(ϑ ′ )ñ sin 2ϕ ′ á F34 (ϑ ′ )ñ sin 2ϕ ′ ú
ê
=
.
ê− á F12 (ϑ ′ )ñ sin 2ϕ ′ − á F22 (ϑ ′ )ñ sin 2ϕ ′ á F33(ϑ ′ )ñ cos 2ϕ ′ á F34 (ϑ ′ )ñ cos 2ϕ ′ú
ê
ú
− á F34 (ϑ ′ )ñ
á F44 (ϑ ′ )ñ
0
0
êë
úû
(4.47)
Inserting this formula and Eqs. (4.32) and (4.40) in Eq. (2.186) yields
98
Scattering, Absorption, and Emission of Light by Small Particles
z
n̂
n̂′
ϑ′
y
ϕ′
x
Figure 4.7. Illustration of the relationship between the phase and scattering matrices when the
scattered light propagates along the positive z-axis.
áΚ e (nˆ , T , ω )ñ ≡ áΚ e (T , ω )ñ = áCabs ñIb (T , ω ),
(4.48)
where áCabs ñ may depend on frequency and Ib (T , ω ) is the blackbody Stokes vector
defined by Eq. (2.184). Thus, the radiation emitted by a small volume element comprising equal numbers of randomly oriented particles and their mirror-symmetric
counterparts and/or randomly oriented particles having a plane of symmetry is not
only isotropic but also unpolarized. The first (and the only non-zero) element of the
average emission vector per particle is simply equal to the product of the average absorption cross section and the Planck function. Substituting Eq. (4.48) in Eq. (2.187),
we see that the emission component of the average radiation force exerted on particles
forming a macroscopically isotropic and mirror-symmetric medium is identically
equal to zero:
áFe (T )ñ ≡ 0.
4.8
Spherically symmetric particles
The structure of the scattering matrix simplifies further for spherically symmetric
particles, that is, for homogeneous or radially inhomogeneous spherical bodies composed of optically isotropic materials. The refractive index inside such particles is a
function of only the distance from the particle center. Irrespective of their “orientation” relative to the laboratory reference frame, spherically symmetric particles are
obviously mirror-symmetric with respect to the xz-plane. Directing the incident light
4 Scattering matrix and isotropic and mirror-symmetric media
99
along the positive z-axis, restricting the scattering direction to the xz-plane with
x ≥ 0, and using this plane for reference, we find from Eqs. (4.2a) and (4.2c) that the
amplitude scattering matrix is always diagonal ( S12 = S 21 = 0). Therefore, Eqs.
(2.106), (2.111), (2.116), (2.121), and (4.7) yield
0
0 ù
é F11(Θ ) F12 (Θ )
ê
ú
0
0 ú
F12 (Θ ) F11(Θ )
ê
.
F(Θ ) =
ê 0
0
F33(Θ ) F34 (Θ )ú
ê
ú
0
− F34 (Θ ) F33(Θ ) úû
êë 0
(4.49)
A scattering matrix of this type appears in the standard Lorenz–Mie theory of light
scattering by homogeneous isotropic spheres; therefore, the above matrix will be referred to as the Lorenz–Mie scattering matrix. The results of the previous sections on
forward and backward scattering imply that
F33(0) = F11(0)
4.9
and
F33(π ) = − F11(π ).
(4.50)
Effects of nonsphericity and orientation
The previous discussion of symmetries enables us to summarize the most fundamental
effects of particle nonsphericity and orientation on single-scattering patterns. If particles
are not spherically symmetric and do not form a macroscopically isotropic and mirrorsymmetric medium, then, in general,
●
●
●
●
●
●
the 4× 4 extinction matrix does not degenerate to a direction- and polarizationindependent scalar extinction cross section;
the extinction, scattering, absorption, and radiation-pressure cross sections, the
single-scattering albedo, and the asymmetry parameter depend on the direction
and polarization state of the incident beam;
all four elements of the emission vector are non-zero and orientation dependent;
the direction of the radiation force does not coincide with the direction of incidence, and the emission component of the radiation force is non-zero;
the scattering matrix F does not have the simple block-diagonal structure of Eq.
(4.7): all 16 elements of the scattering matrix can be non-zero and depend on the
incidence direction and the orientation of the scattering plane rather than only on
the scattering angle;
the phase matrix depends on the specific values of the azimuthal angles of the
incidence and scattering directions rather than on their difference, it cannot be
represented in the form of Eqs. (4.14) and (4.15), and it does not obey the symmetry relations (4.20) and (4.22).
Any of these effects can directly indicate the presence of oriented particles lacking
spherical symmetry. For example, measurements of interstellar polarization are used in
Scattering, Absorption, and Emission of Light by Small Particles
100
astrophysics to detect preferentially oriented dust grains causing different values of extinction for different polarization components of the transmitted starlight (Martin 1978).
Similarly, the depolarization of radiowave signals propagating through the Earth’s atmosphere may indicate the presence of partially aligned nonspherical hydrometeors
(Oguchi 1983).
If nonspherical particles are randomly oriented and form a macroscopically isotropic and mirror-symmetric scattering medium, then
●
●
●
●
●
●
the extinction matrix reduces to the scalar extinction cross section, Eq. (4.32);
all optical cross sections, the single-scattering albedo, and the asymmetry parameter become orientation and polarization independent;
the emitted radiation becomes isotropic and unpolarized;
the radiation force is directed along the incident beam, and the emission component of the radiation force vanishes;
the phase matrix depends only on the difference between the azimuthal angles
of the incidence and scattering directions rather than on their specific values,
has the structure specified by Eqs. (4.14) and (4.15), and obeys the symmetry
relations (4.20) and (4.22);
the scattering matrix becomes block-diagonal (Eq. (4.7)), depends only on the
scattering angle, and possesses almost the same structure as the Lorenz–Mie
scattering matrix (4.49).
However, the remaining key point is that the Lorenz–Mie identities F22 (Θ ) ≡ F11(Θ )
and F44 (Θ ) ≡ F33(Θ ) do not hold, in general, for nonspherical particles. This difference makes measurements of the linear backscattering depolarization ratio δ L =
[ F11(π ) − F22 (π )] [ F11(π ) + F22 (π )] and the closely related circular backscattering
depolarization ratio δ C the most reliable indicators of particle nonsphericity (Sections
10.2 and 10.11). Besides this qualitative distinction, which unequivocally distinguishes randomly oriented nonspherical particles from spheres, there can be significant quantitative differences in specific scattering patterns. They will be discussed in
detail in the following chapters.
4.10
Normalized scattering and phase matrices
It is convenient and customary in many types of applications to use the so-called
normalized scattering matrix
0
0 ù
éa1(Θ ) b1(Θ )
ê
ú
b1(Θ ) a 2 (Θ )
0
0 ú
~
4π
4π
ê
F(Θ ) =
F(Θ ) =
áF(Θ )ñ =
,
ê 0
áCsca ñ
Csca
a3(Θ ) b2 (Θ ) ú
0
ê
ú
− b2 (Θ ) a 4 (Θ )úû
0
êë 0
(4.51)
the elements of which are dimensionless. Similarly, the normalized phase matrix can
4 Scattering matrix and isotropic and mirror-symmetric media
101
be defined as
~
4π
Z(ϑ sca , ϕ sca ; ϑ inc, ϕ inc ) =
Z(ϑ sca , ϕ sca ; ϑ inc, ϕ inc )
Csca
=
4π
á Z(ϑ sca , ϕ sca ; ϑ inc , ϕ inc )ñ.
áCsca ñ
(4.52)
The (1,1) element of the normalized scattering matrix, a1(Θ ), is traditionally called
the phase function and, as follows from Eqs. (4.40) and (4.51), satisfies the normalization condition:
1
2
π
dΘ sinΘ a1(Θ ) = 1.
(4.53)
0
Remember that we have already used the term “phase function” to name the quantity
p defined by Eq. (2.167). It can be easily seen from Eqs. (2.166), (2.167), (4.1), and
(4.51) that the differential scattering cross section dCsca dΩ reduces to á F11 ñ, and so
p reduces to a1 , when unpolarized incident light propagates along the positive z-axis
and is scattered in the xz-plane with x ≥ 0. Equations (4.41) and (4.51) yield
ácosΘ ñ =
1
2
π
dΘ sinΘ a1(Θ ) cosΘ .
(4.54)
0
The normalized scattering matrix possesses many properties of the regular scattering
matrix, e.g.,
a1 ≥ 0,
|ai | ≤ a1,
(4.55)
i = 2, 3, 4,
|bi | ≤ a1,
i = 1, 2,
(4.56)
(a3 + a 4 ) 2 + 4b22 ≤ (a1 + a 2 ) 2 − 4b12 ,
(4.57)
|a3 − a 4 | ≤ a1 − a 2 ,
(4.58)
|a 2 − b1 | ≤ a1 − b1,
(4.59)
|a 2 + b1 | ≤ a1 + b1,
(4.60)
0
0
0 ù
éa1(0)
ê
ú
0
0
0 ú
a 2 (0)
~
,
F ( 0) = ê
ê 0
0
0 ú
a 2 ( 0)
ê
ú
0
0
a 4 (0)úû
êë 0
(4.61)
0
0
0 ù
éa1(π )
ê
ú
a 2 (π )
0
0
0 ú
~
ê
,
F(π ) =
ê 0
− a 2 (π )
0
0 ú
ê
ú
a 4 (π )úû
0
0
êë 0
(4.62)
a 4 (π ) = a1(π ) − 2a 2 (π ),
(4.63)
a 2 (π ) ≥ 0.
102
Scattering, Absorption, and Emission of Light by Small Particles
Also,
a 4 (0) = 2a 2 (0) − a1(0),
0 ≤ a 2 (0) ≤ a1(0)
(4.64)
for rotationally symmetric particles and
0
0 ù
éa1(Θ ) b1(Θ )
ê
ú
b1(Θ ) a1(Θ )
0
0 ú
~
ê
,
F(Θ ) =
ê 0
a3(Θ ) b2 (Θ ) ú
0
ê
ú
− b2 (Θ ) a3(Θ )úû
0
êë 0
a3(0) = a1(0),
a3(π ) = −a1(π )
(4.65)
(4.66)
for spherically symmetric particles. Similarly, for 0 < ϕ sca − ϕ inc < π the normalized
phase matrix is given by
~
Z (ϑ sca , ϕ sca ; ϑ inc , ϕ inc )
C1b1(Θ )
S1b1(Θ )
0 ù
é a1(Θ )
ú
ê
C2b1(Θ )
C1C2 a2(Θ ) − S1S 2 a3(Θ )
S1C2 a2(Θ ) + C1S 2 a3(Θ ) S 2b2(Θ ) ú
ê
=
ê− S 2b1(Θ ) − C1S 2 a2(Θ ) − S1C2 a3(Θ ) − S1S 2 a2(Θ ) + C1C2 a3(Θ ) C2b2(Θ )ú
ú
ê
S1b2(Θ )
a4(Θ ) ûú
0
− C1b2(Θ )
ëê
(4.67)
(cf. Eq. (4.14)) and has the same symmetry properties as the regular phase matrix:
~
~
Z(ϑ sca , ϕ inc; ϑ inc , ϕ sca ) = ∆ 34 Z(ϑ sca , ϕ sca ; ϑ inc , ϕ inc ) ∆ 34 ,
(4.68)
~
~
Z(π − ϑ sca , ϕ sca ; π − ϑ inc , ϕ inc ) = ∆ 34 Z(ϑ sca , ϕ sca ; ϑ inc , ϕ inc ) ∆ 34 ,
(4.69)
~
~
Z(π − ϑ inc , ϕ inc + π ; π − ϑ sca , ϕ sca + π ) = ∆ 3 [Z(ϑ sca , ϕ sca ; ϑ inc , ϕ inc )]T ∆ 3.
(4.70)
An important difference between the regular and normalized matrices is that the
latter do not possess the property of additivity. Consider, for example, a small volume element containing N1 particles of type 1 and N 2 particles of type 2. The total
phase and scattering matrices of the volume element are obtained by adding the phase
and scattering matrices of all particles,
Z = N1 á Z1 ñ + N 2 á Z 2 ñ ,
(4.71)
F = N1 áF1 ñ + N 2 áF2 ñ ,
(4.72)
whereas the respective normalized matrices are given by more complicated relations,
~
~
~ N1 áCsca1 ñ Z1 + N 2 áCsca 2 ñ Z 2
,
(4.73)
Z=
N1 áCsca1 ñ + N 2 áCsca 2 ñ
~
~
~ N áC ñF + N 2 áCsca 2 ñF2
F = 1 sca1 1
(4.74)
N1 áCsca1 ñ + N 2 áCsca 2 ñ
4 Scattering matrix and isotropic and mirror-symmetric media
103
(see Eqs. (4.51) and (4.52)).
4.11
Expansion in generalized spherical functions
A traditional way of specifying the elements of the normalized scattering matrix is to
tabulate their numerical values at a representative grid of scattering angles (Deirmendjian 1969). However, a more mathematically appealing and efficient way is to
expand the scattering matrix elements in so-called generalized spherical functions
s
s
s
(cosΘ ) or, equivalently, in Wigner functions d mn
Pmn
(Θ ) = i n − m Pmn
(cosΘ ) (Siewert
1981; de Haan et al. 1987):
s max
a1(Θ ) =
s max
α
s s
1 P00 (cos
Θ) =
s =0
s
α1s d 00
(Θ ),
(4.75)
s =0
s max
a 2 (Θ ) + a3(Θ ) =
s max
(α + α
s
2
s
s
3 ) P22 (cos
Θ) =
s=2
s
(α 2s + α 3s ) d 22
(Θ ),
s =2
s max
a 2 (Θ ) − a3(Θ ) =
s max
(α − α
s
2
s
s
3 ) P2 , −2 (cos
Θ) =
s=2
s =0
s
α 4s d 00
(Θ ),
(4.78)
s =0
s max
s max
β
s s
1 P02 (cos
Θ) = −
s =2
s
β1s d 02
(Θ ),
(4.79)
s
β 2s d 02
(Θ ).
(4.80)
s=2
s max
b2 (Θ ) =
(4.77)
s max
α 4s P00s (cosΘ ) =
b1(Θ ) =
(α 2s − α 3s ) d 2s, −2 (Θ ),
s =2
s max
a 4 (Θ ) =
(4.76)
s max
β
s s
2 P02 (cos
Θ) = −
s=2
s=2
The number of non-zero terms in the expansions (4.75)–(4.80) is in principle infinite.
In practice, however, the expansions are truncated at s = s max , smax being chosen
such that the corresponding finite sums differ from the respective scattering matrix
elements on the entire interval Θ ∈ [0, π ] of scattering angles within the desired numerical accuracy.
The properties of the generalized spherical functions and the Wigner d-functions
are summarized in Appendix B. For given m and n, either type of function with
s ≥ max(|m|, |n|), when multiplied by
s + 12 , forms a complete orthonormal set of
functions of cos Θ ∈ [−1, + 1] (see Eqs. (B.17) and (B.33)). Therefore, using the orthogonality relation (B.17), we obtain from Eqs. (4.75)–(4.80)
α = (s +
s
1
π
1
)
2
0
s
dΘ sinΘ a1(Θ ) d 00
(Θ ),
(4.81)
104
Scattering, Absorption, and Emission of Light by Small Particles
α 2s + α 3s = ( s + 12 )
s
3
α = (s +
s
4
1
)
2
π
1
)
2
s
dΘ sinΘ [a 2 (Θ ) + a3(Θ )] d 22
(Θ ),
(4.82)
dΘ sinΘ [a 2 (Θ ) − a3(Θ )] d 2s, −2 (Θ ),
(4.83)
0
π
α − α = (s +
s
2
π
0
s
dΘ sinΘ a 4 (Θ ) d 00
(Θ ),
(4.84)
0
β1s = − ( s + 12 )
β 2s = − ( s + 12 )
π
s
dΘ sinΘ b1(Θ ) d 02
(Θ ),
(4.85)
s
dΘ sinΘ b2 (Θ ) d 02
(Θ )
(4.86)
0
π
0
(cf. Eq. (B.21)). These formulas suggest a simple, albeit not always the most elegant
and efficient, way to compute the expansion coefficients by evaluating the integrals
numerically using a suitable quadrature formula (de Rooij and van der Stap 1984). Of
course, this procedure assumes the knowledge of the scattering matrix elements at the
quadrature division points. The expansions (4.75)–(4.80) converge (in the sense of
Eqs. (B.34)–(B.37) or Eqs. (B.18)–(B.21)) to the respective elements of the normalized scattering matrix if these elements are square integrable on the interval
Θ ∈ [0, π ]. In view of the general inequality (4.56), it is sufficient to require that the
phase function a1 (Θ ) be square integrable to ensure such convergence.
Because the Wigner d-functions possess well-known and convenient mathematical
properties and can be efficiently computed by using a simple and numerically stable
recurrence relation, expansions (4.75)–(4.80) offer several practical advantages. First,
we note that according to Eqs. (B.6)–(B.7),
s
d 2s, −2 (0) = d 02
( 0) = 0
(4.87)
s
s
d 22
(π ) = d 02
(π ) = 0.
(4.88)
and
Therefore, Eqs. (4.76), (4.77), (4.79), and (4.80) reproduce identically the specific
structure of the normalized scattering matrix for the exact forward and backward directions, Eqs. (4.61) and (4.62) (cf. Domke 1974). Second, when the expansion coefficients appearing in these expansions are known, then the elements of the normalized
scattering matrix can be calculated easily for practically any number of scattering angles
and with a minimal expenditure of computer time. Thus, instead of tabulating the elements of the scattering matrix for a large number of scattering angles (cf. Deirmendjian
1969) and resorting to interpolation in order to find the scattering matrix at intermediate
points, one can provide a complete and accurate specification of the scattering matrix by
tabulating a limited (and usually small) number of numerically significant expansion
coefficients. This also explains why the expansion coefficients are especially convenient
4 Scattering matrix and isotropic and mirror-symmetric media
105
in ensemble averaging: instead of computing ensemble-averaged scattering matrix elements, one can average a (much) smaller number of expansion coefficients.
An additional advantage of expanding the scattering matrix elements in generalized
spherical functions is that the latter obey an addition theorem and thereby provide an
elegant analytical way of calculating the coefficients in a Fourier decomposition of the
normalized phase matrix (Kuščer and Ribarič 1959; Domke 1974; de Haan et al.
1987). This Fourier decomposition is then used to handle the azimuthal dependence of
the solution of the vector radiative transfer equation efficiently. Another important advantage offered by expansions (4.75)–(4.80) is that the expansion coefficients for certain
types of nonspherical particle can be calculated analytically without computing the scattering matrix itself (Section 5.5).
The expansion coefficients obey the general inequalities
|α sj | ≤ 2s + 1,
j = 1, 2, 3, 4,
| β js | < (2s + 1)
2,
j = 1, 2.
(4.89)
(4.90)
These and other useful inequalities were derived by van der Mee and Hovenier
s
(1990). Since, for each s, d 00
(Θ ) is a Legendre polynomial Ps (cosΘ ), Eq. (4.75) is
also the well-known expansion of the phase function in Legendre polynomials
(Chandrasekhar 1960; Sobolev 1974; van de Hulst 1980). Equation (B.12) gives
0
(Θ ) ≡ 1. Therefore, Eq. (4.81) and the normalization condition (4.53) yield the
d 00
identity
α10 ≡ 1.
(4.91)
Similarly, the average asymmetry parameter, Eq. (4.54), can be expressed as
ácos Θ ñ =
4.12
α11
.
3
(4.92)
Circular-polarization representation
Equations (4.75)–(4.80) become more compact and their origin becomes more transparent if one uses the circular-polarization representation of the Stokes vector (Kuščer
and Ribarič 1959; Domke 1974; Hovenier and van der Mee 1983). We begin by defining the circular components of a transverse electromagnetic wave as
é Eϑ ù
é E+ ù
ê ú = q ê ú,
ë E− û
ëê Eϕ ûú
(4.93)
where
q=
iù
1 é1
ê
ú.
2 ë1 − i û
(4.94)
106
Scattering, Absorption, and Emission of Light by Small Particles
Using Eqs. (2.30) and (4.94), we find that the corresponding circular-polarization
amplitude scattering matrix C is expressed in terms of the regular amplitude scattering matrix as
éC+ +
C=ê
ëêC− +
C+ − ù
ú
C− − ûú
= qSq−1
=
1 é S11 − iS12 + iS 21 + S 22
ê
2 ë S11 − iS12 − iS 21 − S 22
S11 + iS12 + iS 21 − S 22 ù
ú
S11 + iS12 − iS 21 + S 22 û
(4.95)
where the arguments (nˆ sca , nˆ inc ) are omitted for brevity and
1 é 1
ê
2 ë− i
q−1 =
1ù
ú.
iû
The usefulness of the circular electric vector components becomes clear from the
simple formulas
I 2 = E− E+∗ ,
(4.96a)
I 0 = E+ E+∗ ,
(4.96b)
I −0 = E− E−∗ ,
(4.96c)
E+ E−∗ ,
(4.96d)
I −2 =
which follow, after some algebra, from
é Eϑ ù
é E+ ù
−1
ê ú=q ê ú
ë E− û
ëê Eϕ ûú
and Eqs. (1.54) and (1.60). It is easy to verify using the first equality of Eq. (4.95) and
(4.96) that the circular-polarization phase matrix is given by
Z
CP
=
Z CP
pq
éC − − C +∗ +
ê
∗
êC + − C + +
=ê
C C∗
ê −− −+
êC + − C −∗ +
ë
C − + C +∗ +
C + + C +∗ +
C − + C −∗ +
C + + C −∗ +
C − + C +∗ − ù
ú
C + − C +∗ − C + + C +∗ − ú
,
C − − C −∗ − C − + C −∗ − ú
ú
C + − C −∗ − C + + C −∗ − úû
p, q = 2, 0, − 0, − 2.
C − − C +∗ −
(4.97)
Alternatively, it can be found from Eq. (2.123).
Consider now scattering by a macroscopically isotropic and mirror-symmetric
medium. The normalized scattering and phase matrices in the circular-polarization
~
~
representation are defined by analogy with the matrices F and Z:
4π
~
F CP (Θ ) =
á Z CP (ϑ sca = Θ , ϕ sca = 0; ϑ inc = 0, ϕ inc = 0)ñ,
áCsca ñ
(4.98)
4 Scattering matrix and isotropic and mirror-symmetric media
~
4π
Z CP (ϑ sca , ϕ sca ; ϑ inc , ϕ inc ) =
á Z CP (ϑ sca , ϕ sca ; ϑ inc , ϕ inc )ñ,
áCsca ñ
107
(4.99)
where á Z CP (ϑ sca , ϕ sca ; ϑ inc , ϕ inc )ñ is the average circular-polarization phase matrix
per particle. From Eqs. (2.123), (4.51), (1.62), and (1.66) we have
~
~ CP
F CP = F pq
é a 2 + a3
ê
1 êb1 + ib2
=
2 êb1 − ib2
ê
êë a 2 − a3
b1 + ib2
a1 + a 4
a1 − a 4
b1 − ib2
b1 − ib2
a 2 − a3 ù
ú
a1 − a 4 b1 − ib2 ú
,
a1 + a 4 b1 + ib2 ú
ú
b1 + ib2 a 2 + a3 úû
p, q = 2, 0, − 0, − 2.
Obviously, this matrix has several symmetry properties:
~ CP
~
~
F pq
(Θ ) = FqpCP (Θ ) = F−CP
p , − q (Θ ),
~ CP
~
F pp
(Θ ), F pCP
, − p (Θ ) are real,
~
~ CP
∗
F2CP
0 (Θ ) = [ F2, −0 (Θ )] .
(4.100)
(4.101)
(4.102)
(4.103)
~ CP
An elegant and compact way to expand the elements Fpq is to use generalized
s
spherical functions Ppq
:
~ CP
F pq
(Θ ) =
s max
s
g spq Ppq
(cosΘ ),
p, q = 2, 0, − 0, − 2,
(4.104)
s = max(| p|, |q|)
which indicates the rationale for the specific choice of values for the p, q indices for
the circular-polarization phase matrix and the corresponding Stokes vector component
subscripts (Eq. (4.96)). Another justification for this choice of expansion functions
comes from the consideration of certain properties of the rotation group (Domke
1974). The expression for the expansion coefficients g spq follows from Eqs. (4.104)
and (B.37):
g spq
2s + 1
=
2
+1
~ CP
s
d(cosΘ ) F pq
(Θ ) Ppq
(cosΘ ),
p, q = 2, 0, − 0, − 2.
(4.105)
−1
s
Note that for Ppq
(cosΘ ) no distinction is made between p, q = 0 and p, q = − 0. For
s
the values of p and q used here, all functions Ppq
(cosΘ ) are real-valued (see Eq.
(B.30)). Using Eqs. (4.101)–(4.103), (4.105), and (B.31), we derive the following
symmetry relations:
s
g spq = g qp
= g −s p , − q ,
(4.106)
g spp ,
(4.107)
g sp , − p
are real,
s
g 20
= ( g 2s, −0 ) ∗.
(4.108)
Finally, inserting Eq. (4.104) into Eq. (4.100) yields expansions (4.75)–(4.80) with
expansion coefficients
108
Scattering, Absorption, and Emission of Light by Small Particles
s
α1s = g 00
+ g 0s, −0 ,
(4.109)
s
α 2s = g 22
+ g 2s, −2 ,
(4.110)
s
α 3s = g 22
− g 2s, −2 ,
(4.111)
s
α 4s = g 00
− g 0s, −0 ,
(4.112)
s
,
β1s = 2 Re g 02
(4.113)
s
β 2s = 2 Im g 02
.
(4.114)
By analogy with Eq. (4.14) and using Eqs. (1.101) and (4.100), we find for 0 <
ϕ − ϕ inc < π :
sca
~
~
Z CP (ϑ sca , ϕ sca; ϑ inc, ϕ inc) = LCP (−σ 2)F CP(Θ )LCP (π − σ 1)
é(a2 + a3)e −i 2(σ +σ )
ê
1 ê (b1 + ib2)e − i 2σ
= ê
2 (b1 − ib2)e −i 2σ
ê
êë (a2 − a3)ei 2(σ −σ )
1
2
2
(b1 + ib2)e −i 2σ
2
(b1 − ib2)e −i 2σ
2
(a2 − a3)ei 2(σ
1
−σ 2 )
1
a1 + a4
a1 − a4
(b1 − ib2)ei 2σ
1
1
a1 − a4
a1 + a4
(b1 + ib2)ei 2σ
1
1
(b1 − ib2)ei 2σ
2
(b1 + ib2)ei 2σ
2
(a2 + a3)ei 2(σ
1
+σ 2
ù
ú
ú,
ú
ú
)ú
û
(4.115)
where we have omitted the argument Θ in the a’s and b’s. Applying Eq. (2.123) to
Eqs. (4.68)–(4.70) we derive, after some algebra, the supplementary symmetry relations
~
~
Z CP (ϑ sca , ϕ inc; ϑ inc , ϕ sca ) = A∆ 34 A −1Z CP (ϑ sca , ϕ sca ; ϑ inc , ϕ inc ) A∆ 34 A −1
~
= ∆ CP Z CP (ϑ sca , ϕ sca ; ϑ inc , ϕ inc ) ∆ CP ,
(4.116)
~
~
Z CP (π − ϑ sca , ϕ sca ; π − ϑ inc , ϕ inc ) = ∆ CP Z CP (ϑ sca , ϕ sca ; ϑ inc , ϕ inc ) ∆ CP ,
~
~
Z CP (π − ϑ inc , ϕ inc + π ; π − ϑ sca , ϕ sca + π ) = [Z CP (ϑ sca , ϕ sca ; ϑ inc , ϕ inc )]T ,
(4.117)
(4.118)
where
∆ CP
4.13
é0
ê
0
=ê
ê0
ê
êë1
0 0 1ù
ú
0 1 0ú
.
1 0 0ú
ú
0 0 0úû
(4.119)
Radiative transfer equation
For macroscopically isotropic and mirror-symmetric media, the radiative transfer
equation can be significantly simplified:
4 Scattering matrix and isotropic and mirror-symmetric media
109
dI(r; ϑ , ϕ ; ω )
= − I(r; ϑ , ϕ ; ω )
dτ (r, ω )
+
ϖ (r, ω )
4π
2π
+1
d (cosϑ ′ )
−1
~
dϕ ′ Z(r; ϑ , ϑ ′, ϕ − ϕ ′; ω )I(r; ϑ ′, ϕ ′; ω )
0
+ [1 − ϖ (r, ω )] Ib[T (r ), ω ],
(4.120)
where
dτ (r, ω ) = n0 (r )áCext (r, ω )ñ ds
(4.121)
is the optical pathlength element (cf. Eqs. (3.33), (4.32), (4.44), (4.45), (4.48), and
~
(4.52)). By writing the normalized phase matrix in the form Z(r; ϑ , ϑ ′, ϕ − ϕ ′; ω ), we
explicitly indicate that it depends on the difference of the azimuthal angles of the scattering and incident directions rather than on their specific values (Section 4.3). This important property enables an efficient analytical treatment of the azimuthal dependence of
the multiply scattered light, using a Fourier decomposition of the radiative transfer equation (Kuščer and Ribarič 1959; Domke 1974; de Haan et al. 1987). Numerical methods for solving Eq. (4.120) for the plane-parallel geometry are reviewed by Hansen and
Travis (1974).
Equation (4.120) can be further simplified by neglecting polarization and so replacing
the specific intensity vector by its first element (i.e., the radiance) and the normalized
phase matrix by its (1, 1) element (i.e., the phase function):
dI (r; ϑ , ϕ ; ω )
= − I (r; ϑ , ϕ ; ω )
dτ (r, ω )
ϖ (r, ω )
+
4π
2π
+1
d (cos ϑ ′ )
−1
dϕ ′a1(r, Θ , ω ) I (r; ϑ ′, ϕ ′; ω )
0
+ [1 − ϖ (r, ω )] I b[T (r ), ω ],
(4.122)
cosΘ = cosϑ cosϑ ′ + sinϑ sinϑ ′ cos(ϕ − ϕ ′ )
(4.123)
where
(see Eqs. (2.184), (4.17), and (4.67)). Although ignoring the vector nature of light and
replacing the exact vector radiative transfer equation by its approximate scalar counterpart has no rigorous physical justification, this simplification is widely used when the
medium is illuminated by unpolarized light and only the intensity of multiply scattered
light needs to be computed. The scalar approximation gives poor accuracy when the size
of the scattering particles is much smaller than the wavelength (Chandrasekhar 1960;
Mishchenko et al. 1994), but provides acceptable results for particles comparable to and
larger than the wavelength (Hansen 1971). Analytical and numerical solutions of the
110
Scattering, Absorption, and Emission of Light by Small Particles
scalar radiative transfer equation are discussed by Sobolev (1974), van de Hulst (1980),
Lenoble (1985), Yanovitskij (1997), and Thomas and Stamnes (1999).